Generalized Hurwitz polynomials

Mikhail Tyaglov
Institut für Mathematik, Technische Universität Berlin The work was supported by the Sofja Kovalevskaja Research Prize of Alexander von Humboldt Foundation. Email: tyaglov@math.tu-berlin.de

(August 19, 2025)

Abstract

We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only real and simple zeroes. All proofs are given using properties of rational functions mapping the upper half-plane of the complex plane to the lower half-plane. Matrices with self-interlacing spectra and other applications of generalized Hurwitz polynomials are discussed.

Introduction

The polynomials with zeroes in the open left half-plane of the complex plane are called Hurwitz stable, or just stable. They play an important role in several areas of mathematics and engineering such as stability and bifurcation theory, control theory etc. The Hurwitz stable polynomials are well-studied and closely related to the moment problem, spectral theory, operator theory and orthogonal polynomials. The Hurwitz stable polynomials were studied by C. Hermite, E. Routh, A. Stodola, A. Hurwitz [13, 35, 17, 22, 12]. The present paper was motivated by some problems of the bifurcation theory and concerns the most natural (from our point of view) generalization of Hurwitz stable polynomials in the following way.

Let

p(z)=p_{0}(z^{2})+zp_{1}(z^{2})

be a real polynomial, where $p_{0}$ and $p_{1}$ are its even and odd parts, respectively. Consider its associated rational function

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}~.

In the present paper we study polynomials whose associated function $\Phi$ is a so-called R-function, that is, the function mapping the upper half-plane of the complex plane to the lower half-plane. These functions have real, simple, and interlacing zeroes and poles [19, 16]. Such functions play a very important role in spectral theory of self-adjoint operators [37], in the moment problem [2, 1, 23], in the theory of the Stieltjes string [19], in the theory of oscillation operators [10], in the theory of Jacobi and Stieltjes continued fractions [41] etc. In particular, it is very well known [22, 7, 12, 6] that the polynomial $p$ is Hurwitz stable if and only if its associated function $\Phi$ is an R-function with negative zeroes and poles, and $\deg p_{0}\geqslant\deg p_{1}$ . In Section 3 of the present paper, we prove this fact using methods of complex analysis, and then from this fact we obtain all main properties of Hurwitz stable polynomials and the main criteria of the polynomial stability including connections with total positivity and Stieltjes continued fractions. Section 3.2 is devoted to polynomials with zeroes in the closed half-plane of the complex plane. Note that in [3, 20] there was proved that the infinite Hurwitz matrix of a quasi-stable polynomial is totally nonnegative. In Section 3.2, we proved the converse statement, which is new.

In Section 4, we investigate polynomials whose associated function $\Phi$ maps the upper half-plane to the lower half-plane (with $\deg p_{0}\geqslant\deg p_{1}$ ) but has only positive zeroes and poles in contrast with the Hurwitz stable case . We call those polynomials self-interlacing. If $p(z)$ is a self-interlacing polynomial, then it has only simple real zeroes which interlace with zeroes of the polynomial $p(-z)$ . More exactly, if $\lambda_{i}$ , $i=1,\ldots,n$ , are the zeroes of the self-interlacing polynomial $p$ (of type I), then

0<\lambda_{1}<-\lambda_{2}<\lambda_{3}<\ldots<(-1)^{n-1}\lambda_{n}.

This fact is very curious, because the deep structure of self-interlacing polynomials is very close to the structure of the Hurwitz polynomials but the former ones have only real and simple zeroes. The main result that reveals this connection is Theorem 4.8, which says that a polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ is self-interlacing if and only if the polynomial $q(z)=p(-z^{2})-zp_{1}(z^{2})$ is Hurwitz stable. Section 4 is devoted to the comprehensive description of self-interlacing polynomials and includes analogues of the Hurwitz criterion of stability and of the Liénard and Chipart criterion of stability, and some other criteria of self-interlacing. We also give some examples of matrices whose characteristic polynomials are self-interlacing and describe some numerical observations related to the self-interlacing polynomials. Note that there are three R-functions connected to the self-interlacing polynomials. Namely, let $p$ be self-interlacing. Then it has only real and simple zeroes, so its logarithmic derivative $\dfrac{p^{\prime}}{p}$ is an R-function (see e.g. [16]). Moreover, $p$ is self-interlacing if and only if its associated function $\Phi=\dfrac{p_{1}}{p_{0}}$ is an R-function with positive zeroes and poles (Theorem 4.3). Finally, $p$ is self-interlacing if and only if the function $-\dfrac{p(z)}{p(-z)}$ is an R-function (see the proof of Theorem 4.3).

The full generalization of the Hurwitz stable polynomials (in terms of the function $\Phi$ ) is given in Section 5. We describe the polynomials whose associated function $\Phi$ is an R-function without any restriction on the signs of its zeroes and poles. We call such polynomials generalized Hurwitz polynomials. More exactly, the polynomial $p$ of degree $n$ is called generalized Hurwitz of order $k$ (of type I), $0\leqslant k\leqslant\left[\dfrac{n+1}{2}\right]$ , if it has exactly $k$ zeroes in the closed right half-plane, all of which are nonnegative and simple:

0\leqslant\mu_{1}<\mu_{2}<\cdots<\mu_{k},

such that $p(-\mu_{i})\neq 0$ , $i=1,\ldots,k$ , and $p$ has an odd number of zeroes, counting multiplicities, on each interval $(-\mu_{k},-\mu_{k-1}),\ldots,(-\mu_{3},-\mu_{2}),(-\mu_{2},-\mu_{1})$ . Moreover, the number of zeroes of $p$ on the interval $(-\mu_{1},0)$ (if any) is even, counting multiplicities. The other real zeroes lie on the interval $(-\infty,-\mu_{k})$ : an odd number of zeroes, counting multiplicities, when $n=2l$ , and an even number of zeroes, counting multiplicities, when $n=2l+1$ . All nonreal zeroes of $p$ (if any) are located in the open left half-plane of the complex plane.

We prove a generalization of Hurwitz theorem, Theorem 5.6, which, in fact, is an analogue of Hurwitz criterion of stability for the generalized Hurwitz polynomials. We also establish a generalization of the Liénard and Chipart criterion of stability, Theorem 5.8. Note that in [21] G. F. Korsakov made an attempt to prove this generalization of the Liénard and Chipart criterion. However, his methods did not allow him to prove the whole theorem (for details, see Section 5.1) .

We also should note that V. Pivovarchik in [31] describes some aspects of the generalized Hurwitz polynomials, however, in different form, replacing the lower half-plane by the upper half-plane. In particular, he indirectly establishes that the polynomial $p$ is generalized Hurwitz (shifted Hermite-Biehler with symmetry, in Pivovarchik’s terminology) if and only if its associated function $\Phi$ is an R-function. Precisely, the cases considered in [31] are $\deg p_{0}>\deg p_{1}$ and $\deg p_{0}=\deg p_{1}$ with the leading coefficients of $p_{0}$ and $p_{1}$ of the same sign. Neither generalizations of Hurwitz theorem nor generalizations of the Liénard and Chipart criterion, nor the theory of self-interlacing polynomials, nor connections of the generalized Hurwitz polynomials with continued fractions were established in [31]. However, V. Pivovarchik with H. Woracek described [33, 34] a generalization of his shifted Hermite-Biehler polynomials with symmetry (which are generalized Hurwitz polynomials up to rotation of the upper half-plane to the left half-plane) to entire functions. The main result of [33, 34] is an analogue of our Theorem 5.3 for entire functions, provided by two different methods. The method in [34] coincides with that of [31], where as [34] uses the theory of de Branges and Pontryagin spaces. Our proof of Theorem 5.3 differs from the methods used by V. Pivovarchik and H. Woracek and is based on using the properties of R-functions.

It is curious that V. Pivovarchik came to his shifted Hermite-Biehler polynomials and entire functions with symmetry due to his investigation of spectra of quadratic operator pencils [30] and Sturm-Liouville operators whose boundary conditions contain spectral parameter [32, 40]. The last two articles deal with a generalization of the Regge problem, which has a direct relation to the scattering theory. So the distribution of zeroes of the generalized Hurwitz polynomials has a physical interpretation.

Finally, in Section 5.2 we explain how the generalized Hurwitz polynomials can be used in bifurcation theory. Also, we describe how specific zeroes of a real polynomial $p(z,\alpha)$ move with the movement of the parameter $\alpha$ along $\mathbb{R}$ if $p(z,\alpha)$ is generalized Hurwitz for all real $\alpha$ .

1 Auxiliary definitions and theorems

Consider a rational function

R(z)=\dfrac{q(z)}{p(z)},

(1.1)

where $p$ and $q$ are polynomials with complex coefficients

		$\displaystyle p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad$	$\displaystyle a_{0},a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0,$		(1.2)
		$\displaystyle q(z)=b_{0}z^{n}+b_{1}z^{n-1}+\dots+b_{n},\qquad$	$\displaystyle b_{0},b_{1},\dots,b_{n}\in\mathbb{R},$		(1.3)

so $\deg p=n$ and $\deg q\leqslant n$ . If the greatest common divisor of $p$ and $q$ has degree $l$ , $0\leq l\leq n$ , then the rational function $R$ has exactly $r=n-l$ poles.

1.1 Hankel matrices and Hankel minors

Expand the function (1.1) into its Laurent series at $\infty$ :

R(z)=s_{-1}+\dfrac{s_{0}}{z}+\dfrac{s_{1}}{z^{2}}+\dfrac{s_{2}}{z^{3}}+\cdots.

(1.4)

Here $s_{j}=0$ for $j<n-1-m$ and $s_{n-1-m}=\dfrac{b_{0}}{a_{0}}$ , where $m=\deg q$ .

The sequence of coefficients of negative powers of $z$

s_{0},s_{1},s_{2},\ldots

defines the infinite Hankel matrix $S\mathop{{:}{=}}S(R)\mathop{{:}{=}}\|s_{i+j}\|_{i,j=0}^{\infty}$ .

Definition 1.1.

For a given infinite sequence $(s_{j})_{j=0}^{\infty}$ , consider the determinants

D_{j}(S)=\begin{vmatrix}s_{0}&s_{1}&s_{2}&\dots&s_{j-1}\\ s_{1}&s_{2}&s_{3}&\dots&s_{j}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ s_{j-1}&s_{j}&s_{j+1}&\dots&s_{2j-2}\end{vmatrix},\quad j=1,2,3,\dots,

(1.5)

i.e., the leading principal minors of the infinite Hankel matrix $S$ . These determinants are referred to as the Hankel minors or Hankel determinants.

An infinite matrix is said to have finite rank $r$ if all its minors of order greater than $r$ equal zero whereas there exists at least one nonzero minor of order $r$ . Kronecker [24] proved that, for any infinite Hankel matrix, any minor of order $r$ where $r$ is the rank of the matrix, is a multiple of its leading principal minor of order $r$ . This implies the following result.

Theorem 1.2 (Kronecker [24]).

An infinite Hankel matrix $S=\|s_{i+j}\|_{0}^{\infty}$ has finite rank $r$ if and only if

	$\displaystyle D_{r}(S)$	$\displaystyle\neq$	$\displaystyle 0,$		(1.6)
	$\displaystyle D_{j}(S)$	$\displaystyle=$	$\displaystyle 0,\quad\hbox{\rm for all }j>r.$		(1.7)

The following theorem was established by Gantmacher in [12].

Theorem 1.3.

An infinite Hankel matrix $S=\|s_{i+j}\|_{0}^{\infty}$ has finite rank if and only if the sum of the series

R(z)=\dfrac{s_{0}}{z}+\dfrac{s_{1}}{z^{2}}+\dfrac{s_{2}}{z^{3}}+\cdots

is a rational function of $z$ . In this case the rank of the matrix $S$ is equal to the number of poles of the function $R$ .

Theorems 1.2 and 1.3 imply the following: if the greatest common divisor of the polynomials $p$ and $q$ defined in (1.2)–(1.3) has degree $l$ , $0\leqslant l\leqslant m$ , then the formulæ (1.6)–(1.7) hold for $r=n-l$ for the rational function (1.1).

Let $\widehat{D}_{j}(S)$ denote the following determinants

\widehat{D}_{j}(S)=\begin{vmatrix}s_{1}&s_{2}&s_{3}&\dots&s_{j}\\ s_{2}&s_{3}&s_{4}&\dots&s_{j+1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ s_{j}&s_{j+1}&s_{j+2}&\dots&s_{2j-1}\end{vmatrix},\quad j=1,2,3,\dots

(1.8)

With a slight abuse of notation, we will also write $D_{j}(R)\mathop{{:}{=}}D_{j}(S(R))$ and $\widehat{D}_{j}(R)\mathop{{:}{=}}\widehat{D}_{j}(S(R))$ if the matrix $S=S(R)$ is made of the coefficients (1.4) of the function $R$ .

Theorems 1.2 and 1.3 have the following simple corollaries, which will be useful later.

Corollary 1.4 ([16]).

A rational function $R$ represented by the series (1.4) has at most $r$ poles if and only if

\widehat{D}_{j}(R)=0\quad{\rm for}\;\;j=r+1,r+2,\ldots.

Corollary 1.5 ([16]).

A rational function $R$ with exactly $r$ poles represented by the series (1.4) has a pole at the point $0$ if and only if

\widehat{D}_{r-1}(R)\neq 0\quad\text{and}\quad\widehat{D}_{r}(R)=0.

Otherwise, $\widehat{D}_{r}(R)\neq 0$ .

In the sequel, we are also interested in the number of sign changes in the sequences of Hankel minors $D_{j}(R)$ and $\widehat{D}_{j}(R)$ , which we denote by¹¹1Here we set $D_{0}(R)=\widehat{D}_{0}(R)\equiv 1$ . $\operatorname{{\rm V}}(D_{0}(R),D_{1}(R),\ldots,D_{r}(R))$ and $\operatorname{{\rm V}}(\widehat{D}_{0}(R),\widehat{D}_{1}(R),\ldots,\widehat{D}_{r}(R))$ , respectively. These numbers exist, since $R$ is a real function.

In his remarkable work [9], G. Frobenius proved the following fact that allow us to calculate the number of sign changes in the sequence $D_{0}(R),D_{1}(R),\ldots,D_{r}(R)$ in the case when some minors $D_{j}(R)$ are zero.

Theorem 1.6 (Frobenius [9, 11]).

If, for some integers $i$ and $j$ $(0\leq i<j)$ ,

D_{i}(R)\neq 0,\quad D_{i+1}(R)=D_{i+2}(R)=\cdots=D_{i+j}(R)=0,\quad D_{i+j+1}(R)\neq 0,

(1.9)

then the number $\operatorname{{\rm V}}(D_{0}(R),D_{1}(R),D_{2}(R),\ldots,D_{r}(R))$ of Frobenius sign changes should be calculated by assigning signs as follows:

\mathop{\rm sign}\nolimits D_{i+\nu}(R)=(-1)^{\tfrac{\nu(\nu-1)}{2}}\mathop{\rm sign}\nolimits D_{i}(R),\quad\nu=1,2,\ldots,j.

(1.10)

Definition 1.7.

A matrix (finite or infinite) is called totally nonnegative (strictly totally positive) if all its minors are nonnegative (positive).

Definition 1.8.

An infinite matrix of finite rank $r$ is called $m$ -totally nonnegative ( $m$ -strictly totally positive) of rank $r$ , $m\leqslant r$ , if all its minors are nonnegative (positive) up to order $m$ inclusively.

An $r$ -totally nonnegative ( $r$ -strictly totally positive) matrix of rank $r$ is called totally nonnegative (strictly totally positive) of rank $r$ .

Definition 1.9.

An infinite matrix is called positive (nonnegative) definite of rank $r$ if all its leading principal minors are positive (nonnegative) up to order $r$ inclusive.

For the matrix $A$ (finite or infinite), we denote its minor of order $j$ constructed with rows $i_{1},i_{2},\ldots,i_{j}$ and columns $l_{1},l_{2},\ldots,l_{j}$ by

A\begin{pmatrix}i_{1}&i_{2}&\dots&i_{j}\\ l_{1}&l_{2}&\dots&l_{j}\\ \end{pmatrix}.

Definition 1.10 ([10]).

An infinite matrix $A$ of finite rank $r$ is called $m$ -sign regular, $m\leqslant r$ , if all its minors up to order $m$ (inclusively) satisfy the following inequalities:

(-1)^{\sum\limits_{k=1}^{j}i_{k}+\sum\limits_{k=1}^{j}l_{k}}A\begin{pmatrix}i_{1}&i_{2}&\dots&i_{j}\\ l_{1}&l_{2}&\dots&l_{j}\end{pmatrix}>0,\qquad j=1,\ldots,m.

(1.11)

An $r$ -sign regular matrix of rank $r$ is called sign regular of rank $r$ .

In [12] there was proved the following criterion of total positivity of infinite Hankel matrices of finite order.

Theorem 1.11.

An infinite Hankel matrix $S=\|s_{i+j}\|_{0}^{\infty}$ of finite rank $r$ is strictly totally positive if and only if the following inequalities hold:

\begin{split}&D_{j}(S)>0,\\ &\widehat{D}_{j}(S)>0,\end{split}\qquad j=1,\ldots,r.

(1.12)

In [16], the following fact was proved.

Theorem 1.12.

An infinite Hankel matrix $S=\|s_{i+j}\|_{0}^{\infty}$ of finite rank $r$ is sign regular if and only if the following inequalities hold:

\begin{split}&D_{j}(S)>0,\\ &(-1)^{j}\widehat{D}_{j}(S)>0,\end{split}\qquad j=1,\ldots,r.

(1.13)

1.2 Matrices and minors of Hurwitz type

In this section we introduce infinite matrices (matrices of Hurwitz type) associated with the polynomials $p$ and $q$ defined in (1.2)–(1.3) and discuss the Hurwitz formulaæthat connect Hankel matrices with matrices of Hurwitz type.

Definition 1.13.

Given polynomials $p$ and $q$ from (1.2)–(1.3), define the infinite matrix $H(p,q)$ as follows: if $\deg q<\deg p$ , that is, if $b_{0}=0$ , then

H(p,q)=\begin{pmatrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&\dots\\ 0&b_{1}&b_{2}&b_{3}&b_{4}&b_{5}&\dots\\ 0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\dots\\ 0&0&b_{1}&b_{2}&b_{3}&b_{4}&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix},

(1.14)

if $\deg q=\deg p$ , that is, $b_{0}\neq 0$ , then

H(p,q)=\begin{pmatrix}b_{0}&b_{1}&b_{2}&b_{3}&b_{4}&b_{5}&\dots\\ 0&a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&\dots\\ 0&b_{0}&b_{1}&b_{2}&b_{3}&b_{4}&\dots\\ 0&0&a_{0}&a_{1}&a_{2}&a_{3}&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.

(1.15)

The matrix $H(p,q)$ is called an infinite matrix of Hurwitz type. We denote the leading principal minor of $H(p,q)$ of order $j$ , $j=1,2,\ldots$ , by $\eta_{j}(p,q)$ .

Remark 1.14.

The matrix $H(p,q)$ is of infinite rank since its submatrix obtained by deleting the even or odd rows of the original matrix is a triangular infinite matrix with $a_{0}\neq 0$ on the main diagonal.

Together with the infinite matrix $H(p,q)$ , we consider its specific finite submatrices:

Definition 1.15.

Let the polynomials $p$ and $q$ are given by (1.2)–(1.3).

If $\deg q<\deg p=n$ , then we construct the ${2n}\times{2n}$ matrix

\mathcal{H}_{2n}(p,q)=\begin{pmatrix}b_{1}&b_{2}&b_{3}&\dots&b_{n}&0&0&\dots&0&0\\ a_{0}&a_{1}&a_{2}&\dots&a_{n-1}&a_{n}&0&\dots&0&0\\ 0&b_{1}&b_{2}&\dots&b_{n-1}&b_{n}&0&\dots&0&0\\ 0&a_{0}&a_{1}&\dots&a_{n-2}&a_{n-1}&a_{n}&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&a_{0}&a_{1}&a_{2}&\dots&a_{n}&0\\ 0&0&0&\dots&0&b_{1}&b_{2}&\dots&b_{n}&0\\ 0&0&0&\dots&0&a_{0}&a_{1}&\dots&a_{n-1}&a_{n}\\ \end{pmatrix},

(1.16)

if $\deg q=\deg p=n$ , then we construct the $(2n{+}1)\times(2n{+}1)$ matrix

\mathcal{H}_{2n+1}(p,q)=\begin{pmatrix}a_{0}&a_{1}&a_{2}&\dots&a_{n-1}&a_{n}&0&\dots&0&0\\ b_{0}&b_{1}&b_{2}&\dots&b_{n-1}&b_{n}&0&\dots&0&0\\ 0&a_{0}&a_{1}&\dots&a_{n-2}&a_{n-1}&a_{n}&\dots&0&0\\ 0&b_{0}&b_{1}&\dots&b_{n-2}&b_{n-1}&b_{n}&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&a_{0}&a_{1}&a_{2}&\dots&a_{n}&0\\ 0&0&0&\dots&b_{0}&b_{1}&b_{2}&\dots&b_{n}&0\\ 0&0&0&\dots&0&a_{0}&a_{1}&\dots&a_{n-1}&a_{n}\\ \end{pmatrix}.

(1.17)

Both kinds of matrices $\mathcal{H}_{2n}(p,q)$ and $\mathcal{H}_{2n+1}(p,q)$ are called finite matrices of Hurwitz type. The leading principal minors of these matrices will be denoted by²²2That is, $\nabla_{j}(p,q)$ is the leading principal minor of the matrix $\mathcal{H}_{2n}(p,q)$ of order $j$ if $\deg q<\deg p$ . Otherwise (when $\deg q=\deg p$ ), $\nabla_{j}(p,q)$ denotes the leading principal minor of the matrix $\mathcal{H}_{2n+1}(p,q)$ of order $j$ . $\nabla_{j}(p,q)$ .

In his celebrated work [17], A. Hurwitz found relationships between the minors $D_{j}(R)$ and $\widehat{D}_{j}(R)$ defined in (1.5) and (1.8) and the leading principal minors $\eta_{i}(p,q)$ of the matrix $H(p,g)$ (see Definition 1.13).

Lemma 1.16 ([17, 22, 12, 6, 16]).

Let the polynomials $p$ and $q$ be defined in (1.2)–(1.3) and let

R(z)=\dfrac{q(z)}{p(z)}=s_{-1}+\dfrac{s_{0}}{z}+\dfrac{s_{1}}{z^{2}}+\cdots

The following relations hold between the determinants $\eta_{j}(p,q)$ and $D_{j}(R),\widehat{D}_{j}(R)$ defined in (1.5) and (1.8), respectively.

If $\deg q<\deg p$ , then

$\eta_{2j}(p,q)=a_{0}^{2j}D_{j}(R),\quad j=1,2,\ldots;$ (1.18)

$\eta_{2j+1}(p,q)=(-1)^{j}a_{0}^{2j+1}\widehat{D}_{j}(R),\quad j=0,1,2,\ldots$ (1.19)
If $\deg q=\deg p$ , then

$\eta_{2j+1}(p,q)=b_{0}a_{0}^{2j}D_{j}(R),\quad j=0,1,2,\ldots;$ (1.20)

$\eta_{2j}(p,q)=(-1)^{j-1}b_{0}a_{0}^{2j-1}\widehat{D}_{j-1}(R),\quad j=1,2,\ldots$ (1.21)

From (1.14)–(1.17) we have

If $\deg q<\deg p$ , then

$\nabla_{i}(p,q)=a_{0}^{-1}\eta_{i+1}(p,q),\quad i=1,2,\ldots,2n;$ (1.22)
If $\deg q=\deg p$ , then

$\nabla_{i}(p,q)=b_{0}^{-1}\eta_{i+1}(p,q),\quad i=1,2,\ldots,2n+1.$ (1.23)

1.3 Rational functions mapping the upper half-plane to the lower half-plane

All main theorems related to the generalized Hurwitz polynomials are based on properties of the following very important class of rational functions:

Definition 1.17.

A rational function $F$ is called R-function if it maps the upper half-plane of the complex plane to the lower half-plane³³3In [16] these functions are called R-functions of negative type.:

\operatorname{Im}z>0\Rightarrow\operatorname{Im}F(z)<0.

By now, these functions, as well as their meromorphic analogues, have been considered by many authors and have acquired various names. For instance, these functions are called strongly real functions in the monograph [36] due to their property to take real values only for real values of the argument (more general and detailed consideration can be found in [7], see also [16]).

The following theorem provides the most important (for the present study) properties of R-functions. Some parts of this theorem was considered in [29, 22, 7, 12, 4, 6]. An extended version of this theorem can be found (with proof) in [16, Theorem 3.4].

Theorem 1.18.

Let $h$ and $g$ be real polynomials such that $\deg h-1\leqslant\deg g=n$ . For the real rational function

F=\dfrac{h}{g}

with exactly $r\,(\leqslant n)$ poles, the following conditions are equivalent:

$1)$

$F$ is an R-function:

$\operatorname{Im}z>0\Rightarrow\operatorname{Im}F(z)<0;$ (1.24)

2)

The function $F$ can be represented in the form

\displaystyle F(z)=-\alpha z+\beta+\sum^{r}_{j=1}\frac{\gamma_{j}}{z+\omega_{j}},\quad\alpha\geqslant 0,\,\,\,\beta,\omega_{j}\in\mathbb{R},

(1.25)

where

\gamma_{j}=\dfrac{h(\omega_{j})}{g^{\prime}(\omega_{j})}>0,\quad j=1,\ldots,r;

(1.26)

3)

The zeroes of the polynomials $\widetilde{g}(z)$ and $\widetilde{h}(z)$ , where $\widetilde{g}=g/f$ , $\widetilde{h}=h/f$ , $f=\gcd(g,h)$ , are real, simple and interlacing, that is, between any two consecutive zeroes of one of polynomials there is exactly one zero of the other polynomial, counting multiplicity, and

\exists\;\omega\in\mathbb{R}\,:\,\,\,\widetilde{g}(\omega)\widetilde{h}^{\prime}(\omega)-\widetilde{g}^{\prime}(\omega)\widetilde{h}(\omega)<0;

(1.27)

$4)$

The polynomial

$g(z)=\lambda p(z)+\mu q(z)$ (1.28)

has only real zeroes for any real $\lambda$ and $\mu$ , $\lambda^{2}+\mu^{2}\neq 0$ , and the condition (1.27) is satisfied;
$5)$

Let the function $F$ be represented by the series

$F(z)=s_{-2}z+s_{-1}+\frac{s_{0}}{z}+\frac{s_{1}}{z^{2}}+\frac{s_{2}}{z^{3}}+\dots,$ (1.29)

then $s_{-2}\leqslant 0$ and $s_{-1}\in\mathbb{R}$ . The following inequalities hold

$D_{j}(F)>0,\quad j=1,2,\dots,r,$ (1.30)

where determinants $D_{j}(F)$ are defined in (1.5).

Remark 1.19.

If a function $F$ maps the upper half-plane to the upper half-plane⁴⁴4In [16] such functions are called R-functions of positive type., then the function $-F$ is an R-function.

Note that from (1.25) it follows that if $F$ is an R-function, then the function $F$ is decreasing between its poles, so $-F$ is obviously increasing. Therefore, from Theorem 1.18 we obtain the following well-known fact.

Corollary 1.20.

Let $g$ and $h$ be real polynomials such that $|\deg h-\deg g|\leqslant 1$ and let the zeroes of the polynomials $g$ and $h$ be real and simple. If zeroes of $g$ and $h$ interlace, then the function $\dfrac{h}{g}$ is decreasing or increasing between its poles.

Remark 1.21.

If $R$ defined in (1.1) is an R-function, then by Theorem 1.18, we have $\deg q\geqslant\deg p-1$ , that is, the equality $b_{0}=0$ implies $b_{1}\neq 0$ or, in other words, $b_{0}^{2}+b_{1}^{2}\neq 0$ .

In the sequel, we also use the following fact [16], which is a simple consequence of a theorem of V. Markov [7].

Theorem 1.22.

Let $p$ and $q$ be real coprime polynomials such that $|\deg p-\deg q|\leqslant 1$ . If the function $R=q/p$ is an $R$ -function, then the functions $R_{j}=q^{(j)}/p^{(j)}$ , $j=1,\ldots,\deg p-1$ , are also $R$ -functions.

The following simple properties of R-functions can be established using Theorem 1.18.

Theorem 1.23.

Let $F$ and $G$ be R-functions. Then

1)

the function $R=F+G$ is an R-function;
2)

the function $-F(-z)$ is an R-function;
3)

the function $-\dfrac{1}{F(z)}$ is an R-function;
4)

the function $zF(z)$ is an R-function if all poles of $F$ are positive and $F(z)\to 0$ as $z\to\infty$ .

Now consider again the function $R$ defined in (1.1). By (1.2)–(1.3), the degree of its numerator is not greater than the degree of its denominator. If $R$ is an R-function, then one can find the numbers of its negative and positive poles (see e.g. [16]).

Theorem 1.24.

Let a rational function $R$ with exactly $r$ poles be an R-function of negative type and let $R$ have a series expansion (1.29). Then the number $r_{-}$ of negative poles of $R$ equals⁵⁵5Recall that the number $\operatorname{{\rm V}}(1,\widehat{D}_{1}(R),\widehat{D}_{2}(R),\ldots,\widehat{D}_{k}(R))$ of Frobenius sign changes must be calculated according to Frobenius rule provided by Theorem 1.6.

r_{-}=\operatorname{{\rm V}}(1,\widehat{D}_{1}(R),\widehat{D}_{2}(R),\ldots,\widehat{D}_{k}(R)),

(1.31)

where $k=r-1$ , if $R(0)=\infty$ , and $k=r$ , if $|R(0)|<\infty$ . The determinants $\widehat{D}_{j}(R)$ are defined in (1.8).

As was shown in [12] (see also [16]), if $R$ is an R-function has exactly $r$ poles, all of which are of the same sign, then all minors $\widehat{D}_{j}(R)$ are non-zero up to order $r$ .

Corollary 1.25 ([12]).

Let a rational function $R$ with exactly $r$ poles (counting multiplicities) be an R-function of negative type. All poles of $R$ are positive if and only if

\widehat{D}_{j}(R)>0,\qquad j=1,\ldots,r,

where the determinants $\widehat{D}_{j}(R)$ are defined in (1.8).

Corollary 1.26 ([16]).

Let a rational function $R$ with exactly $r$ poles be an R-function of negative type. All poles of $R$ are negative if and only if

(-1)^{j}\widehat{D}_{j}(R)>0,\qquad j=1,\ldots,r,

where the determinants $\widehat{D}_{j}(R)$ are defined in (1.8).

One can use (1.18)–(1.23) to obtain criteria for the function $R$ defined in (1.1) to be an R-function in terms of Hurwitz minors.

Definition 1.27.

For a sequence of real numbers $a_{0},a_{1},\ldots,a_{n}$ , we denote the number of sign changes in the sequence of its nonzero entries by $v(a_{0},a_{1},\ldots,a_{n})$ . The number $v(a_{0},a_{1},\ldots,a_{n})$ is usually called the number of strong sign changes of the sequence $a_{0},a_{1},\ldots,a_{n}$ .

In [16], the following theorem was established.

Theorem 1.28.

Let $R$ be a real rational function as in (1.1). If $R$ is an R-function with exactly $n$ poles, that is, $\gcd(p,q)\equiv 1$ , then the number of its positive poles equals $v(a_{0},a_{1},\ldots,a_{n})$ . In particular, $R$ has only negative poles if and only if ⁶⁶6In fact, the coefficients must be of the same signs, but we assume that $a_{0}>0$ (see (1.2)). $a_{j}>0$ for $j=1,2,\ldots,n$ , and it has only positive poles if and only if $a_{j-1}a_{j}<0$ for $j=1,2,\ldots,n$ .

Remark 1.29.

It is easy to see that Theorem 1.28 is also true for an R-function with degree of numerator greater than degree of denominator.

For R-functions with only negative poles, there are a few more criteria [16].

Theorem 1.30.

The function (1.1), where $\deg q<\deg p$ , is an R-function and has exactly $n$ negative poles if and only if one of the following equivalent conditions holds

$1)$

$a_{n}>0,a_{n-1}>0,\ldots,a_{0}>0,\quad\nabla_{1}(p,q)>0,\nabla_{3}(p,q)>0,\ldots,\nabla_{2n-1}(p,q)>0$ ;
$2)$

$a_{n}>0,b_{n}>0,b_{n-1}>0,\ldots,b_{1}>0,\quad\nabla_{1}(p,q)>0,\nabla_{3}(p,q)>0,\ldots,\nabla_{2n-1}(p,q)>0$ ;
$3)$

$a_{n}>0,a_{n-1}>0,\ldots,a_{0}>0,\quad\nabla_{2}(p,q)>0,\nabla_{4}(p,q)>0,\ldots,\nabla_{2n}(p,q)>0$ ;
$4)$

$a_{n}>0,b_{n}>0,b_{n-1}>0,\ldots,b_{1}>0,\quad\nabla_{2}(p,q)>0,\nabla_{4}(p,q)>0,\ldots,\nabla_{2n}(p,q)>0$ ;

where $\nabla_{i}(p,q)$ are defined in Definition 1.15.

In the case $\deg q=\deg p$ we have similar criteria.

Theorem 1.31.

The function (1.1), where $\deg q(z)=\deg p(z)$ , is an R-function and has exactly $n$ negative poles if and only if one of the following equivalent conditions holds

$1)$

$a_{n}>0,a_{n-1}>0,\ldots,a_{0}>0,\quad\nabla_{2}(p,q)>0,\nabla_{4}(p,q)>0,\ldots,\nabla_{2n}(p,q)>0$ ;
$2)$

$a_{n}>0,b_{n}>0,b_{n-1}>0,\ldots,b_{0}>0,\quad\nabla_{2}(p,q)>0,\nabla_{4}(p,q)>0,\ldots,\nabla_{2n}(p,q)>0$ ;
$3)$

$a_{n}>0,a_{n-1}>0,\ldots,a_{0}>0,\quad\nabla_{1}(p,q)>0,\nabla_{3}(p,q)>0,\ldots,\nabla_{2n+1}(p,q)>0$ ;
$4)$

$a_{n}>0,b_{n}>0,b_{n-1}>0,\ldots,b_{0}>0,\quad\nabla_{1}(p,q)>0,\nabla_{3}(p,q)>0,\ldots,\nabla_{2n+1}(p,q)>0$ ;

where $\nabla_{i}(p,q)$ are defined in Definition 1.15.

At last, we recall a relation between R-functions with nonpositive poles and some properties of Hurwitz matrices [16].

Theorem 1.32 (Total Nonnegativity of the Hurwitz Matrix).

The following are equivalent:

$1)$

The polynomials $p$ and $q$ defined in (1.2)–(1.3) have only nonpositive zeroes (or $q(z)\equiv 0$ ), and the function $R$ defined in (1.1) is either an R-function (or $R(z)\equiv 0$ ).
$2)$

The infinite matrix of Hurwitz type $H(p,q)$ defined in (1.14)–(1.15) is totally nonnegative.

Theorem 1.32 implies the following corollary [16].

Corollary 1.33.

Let the polynomials $p$ and $q$ be defined in (1.2)–(1.3). The following are equivalent:

$1)$

The function $R=\dfrac{p}{q}$ is an R-function with exactly $n$ poles, all of which are negative.
$2)$

The infinite matrix of Hurwitz type $H(p,q)$ defined in (1.14)–(1.15) is totally nonnegative and $\eta_{k+1}(p,q)\neq 0$ .
$3)$

The finite matrix of Hurwitz type $\mathcal{H}_{k}(p,q)$ defined in (1.16)–(1.17) is nonsingular and totally nonnegative. Here $k=2n$ if $\deg q<\deg p$ , and $k=2n+1$ whenever $\deg q=\deg p$ .

The total nonnegativity of the matrices of Hurwitz matrix also implies the following curious result [16].

Theorem 1.34.

Let the polynomials $p$ and $q$ defined in (1.2)–(1.3) have only nonpositive zeroes, and let the function $R=q/p$ be an R-function of negative type. Given any two positive integers⁷⁷7The number $r$ is choosen such that $b_{0}^{2}+b_{r}^{2}\neq 0$ . $r$ and $l$ such that $rl\leqslant n<(l+1)r$ , the polynomials

	$\displaystyle p_{r,l}(z)$	$\displaystyle=$	$\displaystyle a_{0}z^{l}+a_{r}z^{l-1}+a_{2r}z^{l-2}+\ldots+a_{rl},$
	$\displaystyle q_{r,l}(z)$	$\displaystyle=$	$\displaystyle b_{0}z^{l}+b_{r}z^{l-1}+b_{2r}z^{l-2}+\ldots+b_{rl}$

have only negative zeroes, and the function $R_{r,l}=q_{r,l}/p_{r,l}$ is an R-function.

1.4 Stieltjes continued fractions

Let a real rational function $F$ finite at infinity be expanded into its Laurant series at infinity:

F(z)=s_{-1}+\dfrac{s_{0}}{z}+\dfrac{s_{1}}{z^{2}}+\dfrac{s_{2}}{z^{3}}+\cdots

(1.32)

Definition 1.35.

The function $F$ is said to have Stieltjes continued fraction expansion if $F$ can be represented in the form

F(z)=c_{0}+\dfrac{1}{c_{1}z+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}z+\cfrac{1}{\ddots+\cfrac{1}{T}}}}},\quad c_{j}\neq 0,\quad\text{where}\quad T=\begin{cases}&c_{2r}\qquad\ \,\text{if}\ |F(0)|<\infty,\\ &c_{2r-1}z\quad\text{if}\ F(0)=\infty.\end{cases}

(1.33)

Here $r$ is the number of poles of $F$ , counting multiplicity.

There exists the following criterion for a rational function to have a Stieltjes continued fraction expansion [41] (see also [16]).

Theorem 1.36.

Suppose that a rational function $F$ defined in (1.32) has exactly $r$ poles. The function $F$ has a Stieltjes continued fraction expansion (1.33) if and only if it satisfies the conditions

D_{j}(F)\neq 0,\quad j=1,2,\ldots,r,

(1.34)

\widehat{D}_{j}(F)\neq 0,\quad j=1,2,\ldots,r-1,

(1.35)

D_{j}(F)=\widehat{D}_{j}(F)=0,\quad j=r+1,r+2,\ldots,

(1.36)

where $D_{j}(F)$ and $\widehat{D}_{j}(F)$ are defined in (1.5) and (1.8), respectively.

The coefficients of the continued fraction (1.33) can be found by the formulæ

c_{2j}=-\dfrac{D_{j}^{2}(F)}{\widehat{D}_{j-1}(F)\cdot\widehat{D}_{j}(F)},\quad j=1,2,\ldots,r,

(1.37)

c_{2j-1}=\dfrac{\widehat{D}_{j-1}^{2}(F)}{D_{j-1}(F)\cdot D_{j}(F)},\quad j=1,2,\ldots,r,

(1.38)

where $D_{0}(F)=\widehat{D}_{0}(F)\equiv 1$ .

We note that the determinant $\widehat{D}_{r}(F)$ in Theorem 1.36 can be equal zero. But in this case (and only in this case), the function $F$ has a pole at $0$ according to Corollary 1.5, and therefore $T=c_{2r-1}z$ in (1.33).

Let again the polynomials $p$ and $q$ be defined in (1.2)–(1.3). Consider the function $R=\dfrac{q}{p}$ and suppose that $R$ has a Stieltjes continued fraction expansion (1.33), where $r\leqslant n=\deg p$ . Then from (1.18)–(1.21) and (1.37)–(1.38) we obtain:

if $\deg q(z)<\deg p(z)$ , then

	$\displaystyle c_{2j-1}$	$\displaystyle=$	$\displaystyle\dfrac{\eta_{2j-1}^{2}(p,q)}{\eta_{2j-2}(p,q)\cdot\eta_{2j}(p,q)},\qquad j=1,2,\ldots,r;$		(1.39)
	$\displaystyle c_{2j}$	$\displaystyle=$	$\displaystyle\dfrac{\eta_{2j}^{2}(p,q)}{\eta_{2j-1}(p,q)\cdot\eta_{2j}(p,q)},\qquad j=1,2,\ldots,\left[\dfrac{k}{2}\right];$		(1.40)

if $\deg q(z)=\deg p(z)$ , then

	$\displaystyle c_{2j-1}$	$\displaystyle=$	$\displaystyle\dfrac{\eta_{2j}^{2}(p,q)}{\eta_{2j-1}(p,q)\cdot\eta_{2j+1}(p,q)},\qquad j=1,2,\ldots,r;$		(1.41)
	$\displaystyle c_{2j}$	$\displaystyle=$	$\displaystyle\dfrac{\eta_{2j+1}^{2}(p,q)}{\eta_{2j}(p,q)\cdot\eta_{2j+2}(p,q)},\qquad\quad j=0,1,2,\ldots,\left[\dfrac{k}{2}\right];$		(1.42)

where $k=2r-1$ if $R(0)=\infty$ , $k=2r$ if $|R(0)|<\infty$ , and $\eta_{j}(p,q)$ are the leading principal minors of the infinite Hurwitz matrix $H(p,q)$ (see Definition 1.13). Here we set $\eta_{0}(p,q)\equiv 1$ , and $[\rho]$ denotes the largest integer not exceeding $\rho$ .

From Theorems 1.18, 1.24 and 1.36 and from the formulæ (1.37)–(1.38) one obtain the following theorem [16].

Theorem 1.37.

Let the function $R$ with exactly $r$ poles, counting multiplicities, be defined (1.1). If $R$ has a Stieltjes continued fraction expansion (1.33), then $R$ is an R-function if and only if

c_{2j-1}>0,\quad j=1,2,\ldots,r.

(1.43)

Moreover, the number of negative poles of the function $R$ equals the number of positive coefficients $c_{2j}$ , $j=1,2,\ldots,k$ , where $k=r$ , if $|R(0)|<\infty$ , and $k=r-1$ , if $R(0)=\infty$ .

Note that every R-function with poles of the same sign always has a Stieltjes continued fraction expansion.

Corollary 1.38 (A. Markov, Stieltjes, [27, 38, 39]).

A real rational function $R$ with exactly $r$ poles, counting multiplicities, is an R-function with all nonpositive poles if and only if $R$ has a Stieltjes continued fraction expansion (1.33), where

c_{i}>0,\quad i=1,2,\ldots,2r-1.

Moreover, if $|R(0)|<\infty$ , then $c_{2r}>0$ .

Corollary 1.39 ([16]).

A real rational function $R$ with exactly $r$ poles, counting multiplicities, is an R-function with all nonnegative poles if and only if $R$ has a Stieltjes continued fraction expansion (1.33), where

(-1)^{i-1}c_{i}>0,\quad i=1,2,\ldots,2r-1,

and if $|R(0)|<\infty$ , then $c_{2r}<0$ .

2 Even and odd parts of polynomials. Associated function

Consider a real polynomial

p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0.

(2.1)

In the rest of the paper we use the following notation

l=\left[\dfrac{n}{2}\right],

(2.2)

where $n=\deg p$ , and $[\rho]$ denotes the largest integer not exceeding $\rho$ .

The polynomial $p$ can always be represented as follows

p(z)=p_{0}(z^{2})+zp_{1}(z^{2}),

(2.3)

where

for $n=2l$ ,

\begin{split}&p_{0}(u)=a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{n},\\ &p_{1}(u)=a_{1}u^{l-1}+a_{3}u^{l-2}+\ldots+a_{n-1},\end{split}

(2.4)

and for $n=2l+1$ ,

\begin{split}&p_{0}(u)=a_{1}u^{l}+a_{3}u^{l-1}+\ldots+a_{n},\\ &p_{1}(u)=a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{n-1}.\end{split}

(2.5)

The polynomials $p_{0}(z^{2})$ and $p_{1}(z^{2})$ satisfy the following equalities:

\begin{split}&\displaystyle p_{0}(z^{2})=\frac{p(z)+p(-z)}{2},\\ &\displaystyle p_{1}(z^{2})=\frac{p(z)-p(-z)}{2z}.\end{split}

(2.6)

Introduce the following function⁸⁸8In the book [12, Chapter XV], F. Gantmacher used the function $-\dfrac{p_{1}(-u)}{p_{0}(-u)}$ .:

\Phi(u)=\displaystyle\frac{p_{1}(u)}{p_{0}(u)}.

(2.7)

Definition 2.1.

We call $\Phi$ the function associated with the polynomial $p$ .

From (2.6) and (2.7) one can derive the following relations:

z\Phi(z^{2})=\displaystyle\frac{p(z)-p(-z)}{p(z)+p(-z)}=\displaystyle\frac{1-\displaystyle\frac{p(-z)}{p(z)}}{1+\displaystyle\frac{p(-z)}{p(z)}},

(2.8)

\frac{p(-z)}{p(z)}=\displaystyle\frac{1-z\Phi(z^{2})}{1+z\Phi(z^{2})}\,.

Now let us introduce two Hurwitz matrices associated with the polynomial $p$ .

Definition 2.2.

The matrix

H_{\infty}(p)=\begin{pmatrix}a_{0}&a_{2}&a_{4}&a_{6}&a_{8}&a_{10}&\dots\\ 0&a_{1}&a_{3}&a_{5}&a_{7}&a_{9}&\dots\\ 0&a_{0}&a_{2}&a_{4}&a_{6}&a_{8}&\dots\\ 0&0&a_{1}&a_{3}&a_{5}&a_{7}&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}

(2.9)

is called the infinite Hurwitz matrix of the polynomial $p$ . The leading principal minors of the matrix $H_{\infty}(p)$ will be denoted by $\eta_{j}(p)$ , $j=1,2,\ldots$

Remark 2.3.

According to Definitions 1.13 and 2.2, we have $H_{\infty}(p)=H(p_{0},p_{1})$ , where $p_{0}$ and $p_{1}$ are the even and odd parts of the polynomial $p$ , respectively.

Together with the infinite matrix $H_{\infty}(p)$ , we consider its specific finite submatrix.

Definition 2.4.

The $n\times n$ matrix

\mathcal{H}_{n}(p)=\begin{pmatrix}a_{1}&a_{3}&a_{5}&a_{7}&\dots&0&0\\ a_{0}&a_{2}&a_{4}&a_{6}&\dots&0&0\\ 0&a_{1}&a_{3}&a_{5}&\dots&0&0\\ 0&a_{0}&a_{2}&a_{4}&\dots&0&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&\dots&a_{n-1}&0\\ 0&0&0&0&\dots&a_{n-2}&a_{n}\end{pmatrix}

(2.10)

is called the finite Hurwitz matrix or the Hurwitz matrix.

The leading principal minors of this matrix we denote by $\Delta_{j}(p)$ :

\Delta_{j}(p)=\begin{vmatrix}a_{1}&a_{3}&a_{5}&a_{7}&\dots&a_{2j-1}\\ a_{0}&a_{2}&a_{4}&a_{6}&\dots&a_{2j-2}\\ 0&a_{1}&a_{3}&a_{5}&\dots&a_{2j-3}\\ 0&a_{0}&a_{2}&a_{4}&\dots&a_{2j-4}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\dots&a_{j}\end{vmatrix},\quad j=1,\ldots,n,

(2.11)

where we set $a_{i}\equiv 0$ for $i>n$ .

Definition 2.5.

The determinants $\Delta_{j}(p)$ , $j=1,\ldots,n$ , are called the Hurwitz determinants or the Hurwitz minors of the polynomial $p$ .

Remark 2.6.

From Definitions 1.15 and 2.4 it follows that $H_{n}(p)=H_{2l}(p_{0},p_{1})$ if $n=2l$ , and $H_{n}(p)=H_{2l+1}(p_{0},p_{1})$ if $n=2l+1$ , where $p_{0}$ and $p_{1}$ are defined in (2.3)–(2.4).

Obviously,

\eta_{j}(p)=a_{0}\Delta_{j-1}(p),\quad j=1,\dots,n,

(2.12)

where $\Delta_{0}(p)\equiv 1$ .

Suppose that $\deg p_{0}\geqslant\deg p_{1}$ and expand the function $\Phi$ into its Laurent series at infinity:

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=s_{-1}+\frac{s_{0}}{u}+\frac{s_{1}}{u^{2}}+\frac{s_{2}}{u^{3}}+\frac{s_{3}}{u^{4}}+\dots,

(2.13)

where $s_{-1}\neq 0$ if $\deg p_{0}=\deg p_{1}$ , and $s_{-1}=0$ if $\deg p_{0}>\deg p_{1}$ .

From (1.18)–(1.23) we obtain the following relations [17, 12] between the determinants $D_{j}(\Phi)$ , $\widehat{D}_{j}(\Phi)$ defined in (1.5) and (1.8), the Hurwitz minors $\Delta_{j}(p)$ , and the determinants $\eta_{j}(p)$ $(j=1,2,\dots,n)$ .

If $n=2l$ , then

\begin{split}&\Delta_{2j-1}(p)=a_{0}^{-1}\eta_{2j}(p)=a_{0}^{2j-1}D_{j}(\Phi),\\ &\Delta_{2j}(p)=a_{0}^{-1}\eta_{2j+1}(p)=(-1)^{j}a_{0}^{2j}\widehat{D}_{j}(\Phi),\end{split}\qquad j=1,2,\dots,l;

(2.14)

If $n=2l+1$ , then

\begin{split}&\Delta_{2j}(p)=a_{0}^{-1}\eta_{2j+1}(p)={\left(\frac{a_{0}}{s_{-1}}\right)}^{2j}D_{j}(\Phi),\qquad j=1,2,\dots,l;\\ &\Delta_{2j+1}(p)=a_{0}^{-1}\eta_{2j+2}(p)=(-1)^{j}{\left(\frac{a_{0}}{s_{-1}}\right)}^{2j+1}\widehat{D}_{j}(\Phi),\qquad j=0,1,\dots,l,\end{split}

(2.15)

where $\widehat{D}_{0}(\Phi)\equiv 1$ .

By (2.3)–(2.5) and (2.7) and by Theorem 1.2 and Corollary 1.4, we have

D_{j}(\Phi)=\widehat{D}_{j}(\Phi)=0,\quad j>l,

(2.16)

where $l$ is defined in (2.2). So in the sequel, we deal only with the determinants $D_{j}(\Phi)$ , $\widehat{D}_{j}(\Phi)$ of order at most $l$ .

Also in the sequel, we deal only with the determinants $\eta_{j}(p)$ of order at most $n+1$ , since by the formulæ (1.18)–(1.21) and (2.16) and by Remark 2.3, we have

\eta_{j}(p)=0,\quad\text{for}\quad j>n+1.

(2.17)

In Section 5 we also consider the case when $n=2l+1$ with $a_{1}=0$ and $a_{3}\neq 0$ . In this case, we have $\deg p_{0}=\deg p_{1}-1=l-1$ , so the function $\Phi$ has the form

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=s_{-2}u+s_{-1}+\frac{s_{0}}{u}+\frac{s_{1}}{u^{2}}+\frac{s_{2}}{u^{3}}+\frac{s_{3}}{u^{4}}+\dots=\dfrac{a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{2l}}{a_{3}u^{l-1}+a_{5}u^{l-2}+\ldots+a_{2l+1}}.

(2.18)

It is easy to see that $t_{-2}=\dfrac{a_{0}}{a_{3}}$ , so we can represent $\Phi$ as a sum $\Phi(u)=\dfrac{a_{0}}{a_{3}}\,u+\dfrac{p_{2}(u)}{p_{0}(u)}$ , where

p_{2}(u)=\left(a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}\right)u^{l-1}+\left(a_{4}-\dfrac{a_{0}}{a_{3}}\,a_{7}\right)u^{l-2}+\ldots+\left(a_{2l-2}-\dfrac{a_{0}}{a_{3}}\,a_{2l+1}\right)u+a_{2l}.

Note that for the function

\Theta(u):=\dfrac{p_{2}(u)}{p_{0}(u)}=\Phi(u)-\dfrac{a_{0}}{a_{3}}\,u=s_{-1}+\frac{s_{0}}{u}+\frac{s_{1}}{u^{2}}+\frac{s_{2}}{u^{3}}+\frac{s_{3}}{u^{4}}+\dots,

the following equalities hold

D_{j}(\Theta)=D_{j}(\Phi),\quad\text{and}\quad\widehat{D}_{j}(\Theta)=\widehat{D}_{j}(\Phi)\quad\text{for}\quad j=1,2,\ldots

(2.19)

By Definition 1.15 and by the formulæ (1.20), and (1.23) applied to the polynomials $p_{0}$ and $p_{2}$ , we have, for $j=1,2,\ldots$ ,

\begin{array}[]{c}-a_{0}a_{3}^{2j+1}D_{j}(\Theta)=-a_{0}a_{3}\nabla_{2j}(p_{0},p_{2})=-a_{0}a_{3}\begin{vmatrix}a_{3}&a_{5}&\dots&a_{4j+1}\\ a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&a_{4}-\dfrac{a_{0}}{a_{3}}\,a_{7}&\dots&a_{4j}-\dfrac{a_{0}}{a_{3}}\,a_{4j+3}\\ 0&a_{3}&\dots&a_{4j-1}\\ 0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&\dots&a_{4j-2}-\dfrac{a_{0}}{a_{3}}\,a_{4j+1}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&a_{2j+2}-\dfrac{a_{0}}{a_{3}}\,a_{2j+5}\end{vmatrix}=\\ =\begin{vmatrix}0&a_{3}&a_{5}&a_{7}&\dots&a_{4j+3}\\ a_{0}&a_{2}&a_{4}&a_{6}&\dots&a_{4j+2}\\ 0&0&a_{3}&a_{5}&\dots&a_{4j+1}\\ 0&0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&a_{4}-\dfrac{a_{0}}{a_{3}}\,a_{7}&\dots&a_{4j}-\dfrac{a_{0}}{a_{3}}\,a_{4j+3}\\ 0&0&0&a_{3}&\dots&a_{4j-1}\\ 0&0&0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&\dots&a_{4j-2}-\dfrac{a_{0}}{a_{3}}\,a_{4j+1}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\dots&a_{2j+2}-\dfrac{a_{0}}{a_{3}}\,a_{2j+5}\end{vmatrix}=\\ =\begin{vmatrix}0&a_{3}&a_{5}&a_{7}&\dots&a_{4j+3}\\ a_{0}&a_{2}&a_{4}&a_{6}&\dots&a_{4j+2}\\ 0&0&a_{3}&a_{5}&\dots&a_{4j+1}\\ 0&a_{0}&a_{2}&a_{4}&\dots&a_{4j}\\ 0&0&0&a_{3}&\dots&a_{4j-1}\\ 0&0&a_{0}&a_{2}&\dots&a_{4j-2}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\dots&a_{2j+2}\end{vmatrix}=\Delta_{2j+2}(p)=-a_{0}a_{3}^{2j+1}D_{j}(\Phi),\end{array}

where we set $a_{i}\equiv 0$ for $i>2l+1$ .

Denote the coefficients of the polynomial $p_{2}$ by $b_{j}$ :

b_{j}=a_{2j+2}-\dfrac{a_{0}}{a_{3}}a_{2j+5},\qquad j=0,1,\ldots,l-1,

(2.20)

where $a_{2l+3}\equiv 0$ . Then the function $u\Phi(u)$ has the form

u\Phi(u)=\dfrac{a_{0}}{a_{3}}\,u^{2}+\dfrac{b_{0}}{a_{3}}\,u+\dfrac{p_{3}(u)}{p_{0}(u)},

where

p_{3}(u):=up_{2}(u)-\dfrac{b_{0}}{a_{3}}\,up_{0}(u)=\left(b_{1}-\dfrac{b_{0}}{a_{3}}\,a_{5}\right)u^{l-1}+\left(b_{2}-\dfrac{b_{0}}{a_{3}}\,a_{7}\right)u^{l-2}+\ldots+\left(b_{l-1}-\dfrac{b_{0}}{a_{3}}\,a_{2l+1}\right)u.

By Definition 1.15 and by the formulæ (2.19), (1.20) and (1.23) applied to the polynomials $p_{0}$ and $p_{3}$ we get (putting $b_{i}\equiv 0$ for $i>l-1$ ):

\begin{array}[]{c}a_{0}a_{3}^{2j+2}\widehat{D}_{j}(\Theta)=a_{0}a_{3}^{2}\nabla(p_{0},p_{3})=a_{0}a_{3}^{2}\begin{vmatrix}a_{3}&a_{5}&\dots&a_{4j+1}\\ b_{1}-\dfrac{b_{0}}{a_{3}}\,a_{5}&b_{2}-\dfrac{b_{0}}{a_{3}}\,a_{7}&\dots&b_{2j}-\dfrac{b_{0}}{a_{3}}\,a_{4j+3}\\ 0&a_{3}&\dots&a_{4j-1}\\ 0&b_{1}-\dfrac{b_{0}}{a_{3}}\,a_{5}&\dots&b_{2j-1}-\dfrac{b_{0}}{a_{3}}\,a_{4j+1}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&b_{j+1}-\dfrac{b_{0}}{a_{3}}\,a_{2j+5}\end{vmatrix}=\\ =a_{0}a_{3}\begin{vmatrix}a_{3}&a_{5}&a_{7}&\dots&a_{4j+3}\\ 0&a_{3}&a_{5}&\dots&a_{4j+1}\\ 0&b_{1}-\dfrac{b_{0}}{a_{3}}\,a_{5}&b_{2}-\dfrac{b_{0}}{a_{3}}\,a_{7}&\dots&b_{2j}-\dfrac{b_{0}}{a_{3}}\,a_{4j+3}\\ 0&0&a_{3}&\dots&a_{4j-1}\\ 0&0&b_{1}-\dfrac{b_{0}}{a_{3}}\,a_{5}&\dots&b_{2j-1}-\dfrac{b_{0}}{a_{3}}\,a_{4j+1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\dots&b_{j+1}-\dfrac{b_{0}}{a_{3}}\,a_{2j+5}\end{vmatrix}=(-1)^{j}a_{0}a_{3}\begin{vmatrix}a_{3}&a_{5}&a_{7}&\dots&a_{4j+3}\\ b_{0}&b_{1}&b_{2}&\dots&b_{2j}\\ 0&a_{3}&a_{5}&\dots&a_{4j+1}\\ 0&b_{0}&b_{1}&\dots&b_{2j-1}\\ 0&0&a_{3}&\dots&a_{4j-1}\\ 0&0&b_{0}&\dots&b_{2j-2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\dots&a_{2j+3}\end{vmatrix}=\\ =(-1)^{j+1}\begin{vmatrix}0&a_{3}&a_{5}&a_{7}&a_{9}&\dots&a_{4j+5}\\ a_{0}&a_{2}&a_{4}&a_{6}&a_{8}&\dots&a_{4j+4}\\ 0&0&a_{3}&a_{5}&a_{7}&\dots&a_{4j+3}\\ 0&0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&a_{4}-\dfrac{a_{0}}{a_{3}}\,a_{7}&a_{6}-\dfrac{a_{0}}{a_{3}}\,a_{9}&\dots&a_{4j+2}-\dfrac{a_{0}}{a_{3}}\,a_{4j+5}\\ 0&0&0&a_{3}&a_{5}&\dots&a_{4j+1}\\ 0&0&0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&a_{4}-\dfrac{a_{0}}{a_{3}}\,a_{7}&\dots&a_{4j}-\dfrac{a_{0}}{a_{3}}\,a_{4j+3}\\ 0&0&0&0&a_{3}&\dots&a_{4j-1}\\ 0&0&0&0&a_{2}-\dfrac{a_{0}}{a_{3}}\,a_{5}&\dots&a_{4j-2}-\dfrac{a_{0}}{a_{3}}\,a_{4j+1}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&0&\dots&a_{2j+3}\end{vmatrix}=\\ \\ =(-1)^{j+1}\Delta_{2j+3}(p)=a_{0}a_{3}^{2j+2}\widehat{D}_{j}(\Phi),\end{array}

where we set $a_{i}\equiv 0$ for $i>2l+1$ . Here we used the equalities $\widehat{D}_{j}(\Theta)=D_{j}(u\Theta)$ , $j=1,2,\ldots$

Thus, we obtain the following formulæ

\begin{split}&\Delta_{2j}(p)=a_{0}^{-1}\eta_{2j+1}(p)=-a_{0}a_{3}^{2j-1}D_{j-1}(\Phi)=a_{3}^{2j-2}\Delta_{2}(p)D_{j-1}(\Phi),\quad\qquad\qquad\qquad j=2,3,\dots,l,\\ &\Delta_{2j+1}(p)=a_{0}^{-1}\eta_{2j+2}(p)=(-1)^{j}a_{0}a_{3}^{2j}\widehat{D}_{j-1}(\Phi)=(-1)^{j-1}a_{3}^{2j-2}\Delta_{3}(p)\widehat{D}_{j-1}(\Phi),\qquad j=2,3,\dots,l,\end{split}

(2.21)

where $\eta_{i}(p)$ are the leading principal minors of the matrix $H_{\infty}(p)$ defined in (2.9). Here we used the formulæ

\Delta_{2}(p)=a_{0}^{-1}\eta_{3}(p)=-a_{0}a_{3},\quad\Delta_{3}(p)=a_{0}^{-1}\eta_{4}(p)=-a_{0}a_{3}^{2},

(2.22)

that follow from the equality $a_{1}=0$ .

Note that, as above, the following holds:

D_{j}(\Phi)=\widehat{D}_{j}(\Phi)=0,\quad j>l-1.

3 Stable polynomials

This section is devoted to some basic facts of the theory of Hurwitz stable and quasi-stable polynomials. We expound those facts from the viewpoint of the theory of R-functions.

3.1 Hurwitz polynomials

Definition 3.1 ([17, 12]).

The polynomial $p$ defined in (2.1) is called Hurwitz or Hurwitz stable if all its zeroes lie in the open left half-plane of the complex plane.

At first, we recall the following simple necessary condition for polynomials to be Hurwitz stable. This condition is called usually Stodola theorem [12, 6].

Theorem 3.2 (Stodola).

If the polynomial $p$ is Hurwitz stable, then all its coefficients are positive⁹⁹9More exactly, the coefficients must be of the same sign, but $a_{0}>0$ by (2.1)..

Now we show a connection between Hurwitz polynomials and R-functions. This connection will allow us to apply all statements from the Section 1 to Hurwitz polynomials to obtain such basic criteria of Hurwitz stability as Hurwitz criterion and Liénard and Chipart criterion (Theorems (3.5) and 3.7, respectively). We prove the following theorem using a method suggested by Yu.S. Barkovsky [5].

Theorem 3.3 ([12, 6]).

A real polynomial $p$ defined in (2.1) is Hurwitz stable if and only if its associated function $\Phi$ defined in (2.7) is an R-function with exactly $l$ poles, all of which are negative, and the limit $\displaystyle\lim_{u\to\pm\infty}\Phi(u)$ is positive whenever $n=2l+1$ . The number $l$ is defined in (2.2).

Proof.

Let the polynomial $p$ be Hurwitz stable, that is,

p(\lambda)=0\ \ \Longrightarrow\ \ \operatorname{Re}\lambda<0.

(3.1)

First, we show that

\displaystyle\left|\frac{p(-z)}{p(z)}\right|<1,\quad\forall z:\,\operatorname{Re}{z}>0.

(3.2)

Note that the polynomials $p(z)$ and $p(-z)$ have no common zeroes if $p$ is Hurwitz stable, so the function $\dfrac{p(-z)}{p(z)}$ has exactly $n$ poles. The Hurwitz stable polynomial $p$ can be represented in the form

p(z)=a_{0}\prod_{k}(z-\lambda_{k})\prod_{j}\left(z-\xi_{j}\right)\left(z-\overline{\xi}_{j}\right),

where $\lambda_{k}<0,\operatorname{Re}\xi_{j}<0$ and $\operatorname{Im}\xi_{j}\neq 0$ . Then we have

\displaystyle\left|\frac{p(-z)}{p(z)}\right|=\prod_{k}\frac{\left|z+\lambda_{k}\right|}{\left|z-\lambda_{k}\right|}\prod_{j}\frac{\left|z+\xi_{j}\right|\left|z+\overline{\xi}_{j}\right|}{\left|z-\overline{\xi}_{j}\right|\left|z-\xi_{j}\right|}.

(3.3)

It is easy to see that the function of type $\dfrac{z+a}{z-\overline{a}}$ , where $\operatorname{Re}a<0$ , maps the right half-plane of the complex plane to the unit disk. In fact,

\left|\dfrac{z+a}{z-\overline{a}}\right|^{2}=\dfrac{(\operatorname{Re}z+\operatorname{Re}a)^{2}+(\operatorname{Im}z+\operatorname{Im}a)^{2}}{(\operatorname{Re}z-\operatorname{Re}a)^{2}+(\operatorname{Im}z+\operatorname{Im}a)^{2}}<1\qquad\text{whenever}\quad\operatorname{Re}z>0\quad\text{and}\quad\operatorname{Re}a<0.

Now from (3.3) it follows that the function $\displaystyle\frac{p(-z)}{p(z)}$ also maps the right half-plane to the unit disk as a product of functions of such a type. Thus, the inequality (3.2) is valid.

At the same time, the fractional linear transformation $\displaystyle z\mapsto\frac{1-z}{1+z}$ conformally maps the unit disk to the right half-plane:

|z|<1\implies\operatorname{Re}\left(\frac{1-z}{1+z}\right)=\dfrac{1-|z|^{2}}{|1+z|^{2}}>0.

(3.4)

Consequently, from the relations (2.8), (3.2) and (3.4) we obtain that the function $z\Phi(z^{2})$ maps the right half-plane to itself, so the function $-z\Phi(-z^{2})$ maps the upper half-plane of the complex plane to the lower half-plane:

\operatorname{Im}z>0\implies\operatorname{Re}(-iz)>0\implies\operatorname{Re}\left[-iz\Phi(-z^{2})\right]=\operatorname{Im}\left[z\Phi(-z^{2})\right]>0\implies\operatorname{Im}\left[-z\Phi(-z^{2})\right]<0.

Since $p$ is Hurwitz stable by assumption, the polynomials $p(z)$ and $p(-z)$ have no common zeroes, therefore, $p_{0}$ and $p_{1}$ also have no common zeroes, and $p_{0}(0)\neq 0$ by (2.3). Moreover, by Theorem 3.2 we have $a_{0}>0$ and $a_{1}>0$ , so $\deg p_{0}=l$ (see (2.4)–(2.5)). Thus, the number of poles of the function $-z\Phi(-z^{2})$ equals the number of zeroes of the polynomial $p_{0}(-z^{2})$ , i.e. exactly $2l$ .

So according to Theorem 1.18, the function $-z\Phi(-z^{2})$ can be represented in the form (1.25), where all poles are located symmetrically with respect to $0$ and $\beta=0$ , since $-z\Phi(-z^{2})$ is an odd function. Denote the poles of $-z\Phi(-z^{2})$ by $\pm\nu_{1},\ldots,\pm\nu_{l}$ such that

0<\nu_{1}<\nu_{2}<\ldots<\nu_{l}.

Note that $\nu_{1}\neq 0$ , since $p_{0}(0)\neq 0$ as we mentioned above.

Thus, the function $-z\Phi(-z^{2})$ can be represented in the following form

-z\Phi(-z^{2})=-\alpha z+\sum_{j=1}^{l}\frac{\gamma_{j}}{z-\nu_{j}}+\sum_{j=1}^{l}\frac{\gamma_{j}}{z+\nu_{j}}=-\alpha z+\sum_{j=1}^{l}\frac{2\gamma_{j}z}{z^{2}-\nu_{j}^{2}},\quad\alpha\geqslant 0,\,\gamma_{j},\nu_{j}>0.

Dividing this equality by $-z$ and changing variables as follows $-z^{2}\to u$ , $2\gamma_{j}\to\beta_{j}$ , $\nu_{j}^{2}\to\omega_{j}$ , we obtain the following representation of the function $\Phi$ :

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=\alpha+\sum_{j=1}^{l}\frac{\beta_{j}}{u+\omega_{j}},

(3.5)

where $\alpha\geqslant 0,\,\beta_{j}>0$ and

0<\omega_{1}<\omega_{2}<\ldots<\omega_{l}.

Here $\alpha=0$ whenever $n=2l$ , and $\alpha=\displaystyle\frac{a_{0}}{a_{1}}>0$ whenever $n=2l+1$ . Since $\Phi$ can be represented in the form (3.5), we have that by Theorem 1.18, $\Phi$ is an R-function with exactly $l$ poles, all which are negative, and $\displaystyle\lim_{u\to\pm\infty}\Phi(u)=\alpha>0$ as $n=2l+1$ .

Conversely, let the polynomial $p$ be defined in (2.1) and let its associated function $\Phi$ be an R-function with exactly $l$ poles, all of which are negative, and $\displaystyle\lim_{u\to\pm\infty}\Phi(u)>0$ as $n=2l+1$ . We will show that $p$ is Hurwitz stable.

By Theorem 1.18, $\Phi$ can be represented in the form (3.5), where $\alpha=\displaystyle\lim_{u\to\pm\infty}\Phi(u)\geqslant 0$ such that $\alpha>0$ if $n=2l+1$ , and $\alpha=0$ if $n=2l$ . Thus, the polynomial $p_{0}$ has only negative zeroes, and the polynomials $p_{0}$ and $p_{1}$ have no common zeroes. Together with (2.3) and (2.7), this implies that the set of zeroes of the polynomial $p$ coincides with the set of roots of the equation

z\Phi(z^{2})=-1.

(3.6)

Let $\lambda$ be a zero of the polynomial $p$ and therefore, of the equation (3.6). Then from (3.5) and (3.6) we obtain

-1=\operatorname{Re}\left[\lambda\Phi(\lambda^{2})\right]=\left[\alpha+\sum_{j=1}^{l}\beta_{j}\frac{|\lambda|^{2}+\omega_{j}}{|\lambda^{2}+\omega_{j}|^{2}}\right]\operatorname{Re}\lambda,

where $\alpha\geqslant 0$ , and $\beta_{j},\,\omega_{j}>0$ for $j=1,\ldots,l$ . Thus, if $\lambda$ is a zero of $p$ , then $\operatorname{Re}\lambda<0$ , so $p$ is Hurwitz stable. ∎

Consider the Hankel matrix $S(\Phi)=\|s_{i+j}\|^{\infty}_{i,j=0}$ constructed with the coefficients of the series (2.13) and its determinants $D_{j}(\Phi)$ and $\widehat{D}_{j}(\Phi)$ defined in (1.5) and (1.8), respectively. From Theorems 1.18, 3.3 and 1.12 and from Corollary 1.26 we obtain the following Hurwitz stability criteria:

Theorem 3.4.

Let a real polynomial $p$ be defined by (2.1). The following conditions are equivalent:

1)

the polynomial $p$ is Hurwitz stable;

the following hold

\begin{split}&s_{-1}>0\quad\text{for}\quad n=2l+1,\\ &D_{j}(\Phi)>0,\qquad j=1,\ldots,l,\\ &(-1)^{j}\widehat{D}_{j}(\Phi)>0,\qquad j=1,\ldots,l,\end{split}

(3.7)

where $l=\left[\dfrac{n}{2}\right]$ ;

3)

the matrix $S(\Phi)$ is sign regular of rank $l$ , and $s_{-1}>0$ for $n=2l+1$ .

Proof.

In fact, by Theorem 3.3, the polynomial $p$ is Hurwitz stable if and only if its associated function $\Phi$ is an R-function with exactly $l$ poles, all of which are negative and $\lim\limits_{u\to\pm\infty}\Phi(u)=s_{-1}\geqslant 0$ . According to Theorem 1.18 and Corollary 1.26, this is equivalent to the inequalities (3.7). But these inequalities also are equivalent to the sign regularity of the matrix $S(\Phi)$ by Theorem 1.12. ∎

Our next theorem provides stability criteria in terms of coefficients of the polynomial $p$ .

Theorem 3.5.

Given a real polynomial $p$ of degree $n$ as in (2.1), the following conditions are equivalent:

1)

the polynomial $p$ is Hurwitz stable;
2)

all Hurwitz determinants $\Delta_{j}(p)$ are positive:

$\Delta_{1}(p)>0,\ \Delta_{2}(p)>0,\dots,\ \Delta_{n}(p)>0;$ (3.8)
3)

the determinants $\eta_{j}(p)$ are positive up to order $n+1$ :

$\eta_{1}(p)>0,\ \eta_{2}(p)>0,\dots,\ \eta_{n+1}(p)>0;$ (3.9)
4)

the matrix $\mathcal{H}_{n}(p)$ defined in (2.10) is nonsingular and totally nonnegative;
5)

the matrix $H_{\infty}(p)$ defined in (2.9) is totally nonnegative with nonzero minor $\eta_{n+1}(p)$ .

Note that the equivalence of $1)$ and $2)$ is the famous Hurwitz criterion of stability. The implications $1)\Longrightarrow 4)$ and $1)\Longrightarrow 5)$ were proved in [3, 20]. The implication $4)\Longrightarrow 1)$ was, in fact, proved in [3]. However, the implication $5)\Longrightarrow 1)$ is probably new.

Proof.

Indeed, by Theorem 3.4, the polynomial $p$ is Hurwitz stable if and only if the inequalities (3.7) hold. According to the formulæ (2.14)–(2.15), these inequalities are equivalent to (3.8). By (2.12), the inequalities (3.8) are equivalent to (3.9), since $\eta_{1}(p)=a_{0}>0$ (see (1.14)–(1.15) and Remark 2.3).

Furthermore, by Theorem 3.3, the polynomial $p$ is Hurwitz stable if and only if its associated function $\Phi$ is an R-function with exactly $l=\left[\dfrac{n}{2}\right]$ , all of which are negative, and $\lim\limits_{u\to\pm\infty}\Phi(u)=\dfrac{a_{0}}{a_{1}}=s_{-1}>0$ if $n=2l+1$ . According to Corollary 1.33 and Remark 2.3, this is equivalent to the total nonnegativity of the matrix $H_{\infty}(p)$ . Moreover, $\eta_{n+1}(p)\neq 0$ by Corollary 1.33. Thus, the condition $1)$ is equivalent to the condition $5)$ .

The conditions $1)$ and $4)$ are equivalent also by Corollary 1.33 and Remark 2.6. ∎

Remark 3.6.

Note that total nonnegativity of the Hurwitz matrix of a Hurwitz polynomial was established in [3, 20, 14].

Now we are in a position to prove a few another famous criteria of Hurwitz stability, which are known as the Liénard and Chipart criterion and its modifications (see [25, 12]). These criteria are simple consequences of Theorems 1.30, 1.31 and 3.3.

Theorem 3.7.

The polynomial $p$ given by (2.1) is Hurwitz stable if and only if one of the following conditions holds

$1)$

$a_{n}>0,a_{n-2}>0,a_{n-4}>0,\ldots,\qquad\Delta_{n-1}(p)>0,\Delta_{n-3}(p)>0,\Delta_{n-5}(p)>0,\ldots$ ;
$2)$

$a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots,\qquad\Delta_{n-1}(p)>0,\Delta_{n-3}(p)>0,\Delta_{n-5}(p)>0,\ldots$ ;
$3)$

$a_{n}>0,a_{n-2}>0,a_{n-4}>0,\ldots,\qquad\Delta_{n}(p)>0,\Delta_{n-2}(p)>0,\Delta_{n-4}(p)>0,\ldots$ ;
$4)$

$a_{n}>0,a_{n-1}>0,a_{n-3}>0,\ldots,\qquad\Delta_{n}(p)>0,\Delta_{n-2}(p)>0,\Delta_{n-4}(p)>0,\ldots$

Proof.

Let $n=2l$ . By Definitions 1.15 and 2.4 and by Remark 2.6, we have $H_{n}(p)=H_{2l}(p_{0},p_{1})$ , where the polynomials $p_{0}$ and $p_{1}$ are defined in (2.3)–(2.4). Obviously, $\Delta_{j}(p)=\Delta_{j}(p_{0},p_{1})$ , $j=1,\ldots,n$ . The assertion of the theorem follows now from Theorems 3.3 and 1.30.

In the case $n=2l+1$ , the assertion of the theorem can be proved in the same way using Theorem 1.31 instead of Theorem 1.30. ∎

We now recall a connection between Hurwitz polynomials and Stieltjes continued fractions (see [12]).

Theorem 3.8.

The polynomial $p$ of degree $n\geqslant 1$ defined in (2.1) is Hurwitz stable if and only if its associated function $\Phi$ has the following Stieltjes continued fraction expansion:

\Phi(u)=c_{0}+\dfrac{1}{c_{1}u+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}u+\cfrac{1}{\ddots+\cfrac{1}{c_{2l-1}u+\cfrac{1}{c_{2l}}}}}}},\quad\text{with}\quad c_{i}>0,\quad i=1,\ldots,2l,

(3.10)

where $c_{0}=0$ if $n$ is even, $c_{0}>0$ if $n$ is odd, and $l$ as in (2.2).

Proof.

In fact, by Theorem 3.3, the polynomial $p$ is Hurwitz stable if and only if the function $\Phi$ can be represented in the form (3.5), where $\alpha=c_{0}=0$ if $n=2l$ , and $\alpha=c_{0}=\dfrac{a_{0}}{a_{1}}>0$ if $n=2l+1$ . Now the assertion of the theorem follows from Theorem 1.18 and Corollary 1.38. ∎

From the formulæ (1.37)–(1.38) it follows that the coefficients $c_{i}$ of (3.10) can be found by the following formulæ

c_{2j}=-\dfrac{D_{j}^{2}(\Phi)}{\widehat{D}_{j-1}(\Phi)\cdot\widehat{D}_{j}(\Phi)},\quad j=1,2,\ldots,l,

(3.11)

c_{2j-1}=\dfrac{\widehat{D}_{j-1}^{2}(\Phi)}{D_{j-1}(\Phi)\cdot D_{j}(\Phi)},\quad j=1,2,\ldots,l,

(3.12)

where $D_{0}(\Phi)=\widehat{D}_{0}(\Phi)\equiv 1$ . If $n$ is odd, then $c_{0}=\dfrac{a_{0}}{a_{1}}>0$ .

Using the formulæ (3.11)–(3.12) and (2.14)–(2.15), we can represent the coefficients $c_{i}$ in terms of Hurwitz determinants $\Delta_{i}(p)$ :

1)

If $n=2l$ , then

$c_{i}=\dfrac{\Delta^{2}_{i-1}(p)}{\Delta_{i-2}(p)\cdot\Delta_{i}(p)},\quad i=1,2,\dots,n,$ (3.13)
2)

If $n=2l+1$ , then

$c_{i}=\dfrac{\Delta^{2}_{i}(p)}{\Delta_{i-1}(p)\cdot\Delta_{i+1}(p)},\quad i=0,1,2,\dots,n,$ (3.14)

where we set $\Delta_{-1}(p)\equiv\dfrac{1}{a_{0}}$ and $\Delta_{0}(p)\equiv 1$ .

At last, from Theorems 1.22 and 3.3 one can obtain the following simple result.

Theorem 3.9.

Let $p$ be a Hurwitz polynomial of degree $n\geqslant 2$ as in (2.1). Then all the polynomials

p_{j}(z)=\sum\limits_{i=0}^{n-2j}\left[\dfrac{n-i}{2}\right]\left(\left[\dfrac{n-i}{2}\right]-1\right)\cdots\left(\left[\dfrac{n-i}{2}\right]+j-1\right)a_{i}z^{n-2j-i},\quad j=1,\ldots,\left[\dfrac{n}{2}\right]-1,

also are Hurwitz stable.

Proof.

By Theorem 3.3, if $p$ is Hurwitz stable, then the function $\Phi=p_{1}/p_{0}$ is an R-function. According to Theorem 1.22, all functions $\Phi_{j}=p_{1}^{(j)}/p_{0}^{(j)}$ , $j=1,\ldots,\left[\dfrac{n}{2}\right]-1$ , are R-functions with negative zeroes and poles. Now Theorem 3.3 implies that all polynomials

p_{j}(z)=p_{0}^{(j)}(z^{2})+zp_{1}^{(j)}(z^{2}),\quad j=1,\ldots,\left[\dfrac{n}{2}\right]-1,

are Hurwitz stable, as required. ∎

3.2 Quasi-stable polynomials

In this section we deal with polynomials whose zeroes lie in the closed left half-plane of the complex plane.

Definition 3.10.

A polynomial $p$ of degree $n$ is called quasi-stable with degeneracy index $m$ , $0\leqslant m\leqslant n$ , if all its zeroes lie in the closed left half-plane of the complex plane and the number of zeroes of $p$ , counting multiplicities, on the imaginary axis equals $m$ . We call the number $n-m$ the stability index of the polynomial $p$ .

Throughout this section we use the following notation

r=\left[\dfrac{n}{2}\right]-\left[\dfrac{m}{2}\right],

(3.15)

where $n$ and $m$ are degree and degeneracy index of the polynomial $p$ , respectively.

Obviously, any Hurwitz polynomial is quasi-stable with zero degeneracy index, that is, it has the smallest degeneracy index and the largest stability index (which equals the degree of the polynomial). Note that if the degeneracy index $m$ is even, then $p(0)\neq 0$ , and if $m$ is odd, then $p$ must have a zero at $0$ .

Moreover, if $p$ is a quasi-stable polynomial, then

p(z)=p_{0}(z^{2})+zp_{1}(z^{2})=f(z^{2})q(z)=f(z^{2})\left[q_{0}(z^{2})+zq_{1}(z^{2})\right],

where $f(u)$ is a real polynomial of degree $\left[\dfrac{m}{2}\right]$ with nonpositive zeroes, and $q$ is a Hurwitz stable polynomial if $m$ is even, and it is a quasi-stable polynomial with degeneracy index $1$ if $m$ is odd. Using this representation of quasi-stable polynomials, one can extend almost all results of Section 3.1 to quasi-stable polynomials in the same way, so we state them here without proofs. We only should take into account that the function

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=\dfrac{f(u)q_{1}(u)}{f(u)q_{0}(u)}=\dfrac{q_{1}(u)}{q_{0}(u)}

(3.16)

has a pole at $0$ whenever $p(0)=0$ .

The analogue of Stodola necessary condition for quasi-stable polynomials is the following.

Theorem 3.11.

If the polynomial $p$ defined in (2.1) is quasi-stable, then all its coefficients are nonnegative.

The next theorem is an extended version of Theorem 3.3.

Theorem 3.12.

The polynomial $p$ of degree $n$ defined in (2.1) is quasi-stable with degeneracy index $m$ if and only if the function $\Phi$ defined in (3.16) is an R-function of negative type with exactly $r$ poles all of which are nonpositive, and $\displaystyle\lim_{u\to\pm\infty}\Phi(u)$ is positive if $n=2l+1$ . The number $r$ is defined in (3.15).

If we expand the function $\Phi$ into its Laurent series (2.13), then the Hankel matrix $S(\Phi)=\|s_{i+j}\|^{\infty}_{0}$ has rank $r$ , where $r$ as in (3.15).

Theorem 3.13.

Let the polynomial $p$ be defined in (2.1). The following conditions are equivalent:

1)

the polynomial $p$ is quasi-stable with even degeneracy index $m$ ;
2)

the following hold

$s_{-1}>0\quad\text{for}\quad n=2l+1,$

$D_{j}(\Phi)>0,\qquad j=1,\ldots,r,$

$(-1)^{j}\widehat{D}_{j}(\Phi)>0,\qquad j=1,\ldots,r-1,$

and $(-1)^{r}\widehat{D}_{r}(\Phi)>0$ for even $m$ , but $\widehat{D}_{r}(\Phi)=0$ for odd $m$ . Here $r$ is defined in (3.15);
3)

the matrix $S(\Phi)$ is sign regular of rank $r$ for even $m$ , $S(\Phi)$ is $r-1$ -sign regular of rank $r$ for odd $m$ , and $s_{-1}$ is positive if $n=2l+1$ .

It is also easy to extend Theorem 3.5 to quasi-stable polynomials.

Theorem 3.14.

Let the polynomial $p$ of degree $n$ be defined in (2.1). The following conditions are equivalent:

1)

the polynomial $p$ is quasi-stable with degeneracy index $m$ ;
2)

determinants $\Delta_{j}(p)$ are positive up to order $n-m$ :

$\Delta_{1}(p)>0,\ \Delta_{2}(p)>0,\dots,\ \Delta_{n-m}(p)>0,\ \Delta_{n-m+1}(p)=\ldots=\Delta_{n}(p)=0;$
3)

determinants $\eta_{j}(p)$ are positive up to order $n-m+1$ :

$\eta_{1}(p)>0,\ \eta_{2}(p)>0,\dots,\ \eta_{n-m+1}(p)>0,\ \eta_{n-m+i}(p)=0,\quad i=2,3,\ldots;$
4)

the matrix $H_{\infty}(p)$ is totally nonnegative and

$\eta_{n-m+1}(p)\neq 0,\,\eta_{n-m+i}(p)=0,\quad i=2,3,\ldots$

The implication $1)\Longrightarrow 4)$ was proved, indeed, in [3, 20]. The implication $4)\Longrightarrow 1)$ seems to be new. Thus, we established that a polynomial is quasi-stable if and only if its infinite Hurwitz matrix is totally nonnegative.

The following interesting property of quasi-stable polynomials is a simple consequence of Theorems 1.34 and 3.3.

Theorem 3.15.

Let the polynomials $p$ of degree $n=2l$ defined in (2.1) be quasi-stable. Given any positive integer $r(\leqslant n)$ , the polynomial

p_{r}(z)=a_{0}z^{2k}+a_{2r-1}z^{2k-1}+a_{2r}z^{2k-2}+a_{4r-1}z^{2k-3}+a_{4r}z^{2k-4}+\ldots+a_{2rk-1}z+a_{2rk},

where $k=\left[\dfrac{n}{r}\right]$ , also is quasi-stable.

Theorem 3.16.

Let the polynomials $p$ of degree $n=2l+1$ defined in (2.1) be quasi-stable. Given any positive integer $r(\leqslant n)$ , the polynomial

p_{r}(z)=a_{0}z^{2k+1}+a_{1}z^{2k}+a_{2r}z^{2k-1}+a_{2r+1}z^{2k-2}+a_{4r}z^{2k-3}+a_{4r+1}z^{2k-4}+\ldots+a_{2rk}z+a_{2rk+1},

where $k=\left[\dfrac{n}{r}\right]$ , also is quasi-stable.

Although total nonnegativity of the infinite Hurwitz matrix is equivalent to quasi-stability of polynomials, total nonnegativity of the finite singular Hurwitz matrix is not equivalent to quasi-stability as was noticed by Asner [3]. It is clear from Theorem 3.14 that the finite Hurwitz matrix of a quasi-stable polynomial is totally nonnegative. However, given a real polynomial $p$ of degree $n$ , if $\mathcal{H}_{n}(p)$ is totally nonnegative, $p$ is not undertaken to be quasi-stable. In fact [16], $p$ can be represented in the form $p(z)=q(z)g(z^{2})$ , where the polynomial $q$ is Hurwitz stable, and the polynomial $g$ is chosen such that the matrix $\mathcal{H}_{n}(p)$ is totally nonnegative but $g(z^{2})$ has zeroes in the open right half-plane. Such choice is possible as was mentioned in [16].

The connection between quasi-stable polynomials and Stieltjes continued fractions is similar to Hurwitz stable polynomials’ one. Namely, one can prove the following extended version of Theorem 3.8

Theorem 3.17.

The polynomial $p$ of degree $n$ defined in (2.1) is quasi-stable with degeneracy index $m$ if and only if the function $\Phi$ has a Stieltjes continued fraction expansion:

\Phi(u)=c_{0}+\dfrac{1}{c_{1}u+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}u+\cfrac{1}{\ddots+\cfrac{1}{T}}}}},\quad\text{with}\quad c_{i}>0,\quad T=\begin{cases}&c_{2r},\ \text{if}\ m\ \text{is even},\\ &c_{2r-1}u,\ \text{if}\ m\ \text{is odd}.\end{cases}

(3.17)

Here $c_{0}=0$ if $n$ is even, $c_{0}>0$ if $n$ is odd, and $r$ is defined in (3.15).

The coefficients $c_{i}$ in (3.17) also can be found by the formulæ (3.13)–(3.14).

Finally, we note that a quasi-stable polynomial $p$ with degeneracy index $1$ can be represented as follows: $p(z)=zq(z)$ , where $q$ is a Hurwitz stable polynomial. Therefore, for quasi-stable polynomials with degeneracy index $1$ , one can establish an analogue of Liénard and Chipart criterion.

Theorem 3.18.

The polynomial $p$ given by (2.1) is quasi-stable with degeneracy index $1$ if and only if one of the following conditions holds

$1)$

$a_{n}=0,a_{n-1}>0,a_{n-2}>0,a_{n-4}>0,\ldots,\quad\Delta_{1}(p)>0,\Delta_{3}(p)>0,\Delta_{5}(p)>0,\ldots$ ;
$2)$

$a_{n}=0,a_{n-1}>0,a_{n-3}>0,a_{n-5}>0,\ldots,\quad\Delta_{1}(p)>0,\Delta_{3}(p)>0,\Delta_{5}(p)>0,\ldots$ ;
$3)$

$a_{n}=0,a_{n-1}>0,a_{n-2}>0,a_{n-4}>0,\ldots,\quad\Delta_{2}(p)>0,\Delta_{4}(p)>0,\Delta_{6}(p)>0,\ldots$ ;
$4)$

$a_{n}=0,a_{n-1}>0,a_{n-3}>0,a_{n-5}>0,\ldots,\quad\Delta_{2}(p)>0,\Delta_{4}(p)>0,\Delta_{6}(p)>0,\ldots$

4 Self-interlacing polynomials

In Section 3 we established that the function $\Phi$ (see (2.7)) associated with a Hurwitz stable polynomial (quasi-stable polynomial) maps the upper half-plane of the complex plane to the lower half-plane and possesses only negative (nonpositive) poles. Now we are in a position to describe polynomials whose associated function $\Phi$ also maps the upper half-plane to the lower half-plane but has only positive (nonnegative) poles.

4.1 General theory

Definition 4.1.

A real polynomial $p(z)$ is called self-interlacing if all its zeroes are real and simple and interlace zeroes of the polynomial $p(-z)$ .

In other words, if $\lambda_{i}$ are the zeroes of a self-interlacing polynomial $p$ , then one of the following holds:

0<\lambda_{1}<-\lambda_{2}<\lambda_{3}<\ldots<(-1)^{n-1}\lambda_{n},

(4.1)

0<-\lambda_{1}<\lambda_{2}<-\lambda_{3}<\ldots<(-1)^{n}\lambda_{n},

(4.2)

where $n=\deg p$ .

Definition 4.2.

A real polynomial $p$ of degree $n$ is called self-interlacing of type I (of type II) if all its zeroes are real and simple and satisfy the inequalities (4.1) (respectively, (4.2)).

If a polynomial $p$ of degree $n$ is self-interlacing of type I, then its minimal absolute value zero is positive. Moreover, if $n=2l$ , then $p$ has exactly $l$ negative zeroes and exactly $l$ positive zeroes. Its zero $\lambda_{n}$ , which has the maximal absolute value is negative. If $n=2l+1$ , then the polynomial $p$ has exactly $l$ negative zeroes and $l+1$ positive zeroes. In this case, $\lambda_{n}$ is positive.

Note that a polynomial $p(z)$ is self-interlacing of type I if and only if the polynomial $p(-z)$ is self-interlacing of type II. So in the sequel, we deal only with self-interlacing polynomials of type I.

Theorem 4.3.

Let $p$ be a real polynomial of degree $n\geqslant 1$ as in (2.1). The polynomial $p$ is self-interlacing of type I if and only if its associated function $\Phi$ defined in (2.7) is an R-function with exactly $l$ poles, all of which are positive, and $\displaystyle\lim_{u\to\pm\infty}\Phi(u)$ is negative whenever $n=2l+1$ .

Proof.

Let $p$ be self-interlacing of type I. First, we show that the function

G(z)=-\dfrac{p(-z)}{p(z)}

is an R-function. In fact, by Definition 4.1, the zeroes of the polynomials $p(z)$ and $p(-z)$ are real, simple and interlacing, that is, between any two consecutive zeroes of one polynomial there lies exactly one zero, counting multiplicity, of the other polynomial, so $G(z)$ or $-G(z)$ is an R-function according to Theorem 1.18. By Corollary 1.20, $G$ is monotone between its poles. So it remains to prove that the function $G$ is decreasing between its poles.

Let $n=2l$ , then by (4.1), the maximal pole of $G$ is $\lambda_{n-1}>0$ , but its maximal zero is $-\lambda_{n}>0$ , which is greater than $\lambda_{n-1}$ according to Definition 4.2 (see (4.1)). Therefore, in the interval $(-\lambda_{n},+\infty)$ , the function $G$ has no poles and zeroes. At the same time, $\displaystyle\lim_{z\to\pm\infty}G(z)=-1$ . Consequently, in the interval $(-\lambda_{n},+\infty)$ , $G(z)$ decreases from $0$ to $-1$ . So, $G$ is an R-function. In the same way, one can prove that if $n=2l+1$ , then $G$ is also an R-function.

Thus, if $p$ is self-interlacing of type I, then $G$ maps the upper half-plane to the lower half-plane. From (2.8) we obtain that

z\Phi(z^{2})=\dfrac{1+G(z)}{1-G(z)}.

(4.3)

Since the fractional linear transformation $\displaystyle z\mapsto\frac{1+z}{1-z}$ conformally maps the lower half-plane to the lower half-plane:

\operatorname{Im}z<0\implies\operatorname{Im}\left(\frac{1-z}{1+z}\right)=\dfrac{2\operatorname{Im}z}{|1+z|^{2}}<0,

(4.4)

from (4.3)–(4.4) it follows that the function $z\Phi(z^{2})$ maps the upper half-plane to the lower half-plane, that is, $z\Phi(z^{2})$ is an R-function.

Since $p$ is self-interlacing of type I by assumption, the polynomials $p(z)$ and $p(-z)$ have no common zeroes, therefore $p_{0}$ and $p_{1}$ also have no common zeroes and $p_{0}(0)\neq 0$ by (2.3). Therefore, the number of poles of the function $z\Phi(z^{2})$ equals the number of zeroes of the polynomial $p_{0}(z^{2})$ . If $n=2l$ , then by (2.4), $\deg p_{0}=l$ , so $z\Phi(z^{2})$ has exactly $2l$ poles. If $n=2l+1$ , then by (2.5), $\deg p_{1}=l$ and $\deg p_{0}\leqslant l$ , so $z\Phi(z^{2})$ has at most $2l$ poles, and it has exactly $2l+1$ zeroes, since $p_{0}(0)\neq 0$ as we mentioned above. But $z\Phi(z^{2})$ is an R-function, therefore, it has exactly $2l$ poles by Theorem 1.18.

Thus, Theorem 1.18 implies that the function $z\Phi(z^{2})$ can be represented in the form (1.25), where all poles are located symmetrically with respect to $0$ and $\beta=0$ , since $z\Phi(z^{2})$ is an odd function. Denote the poles of $z\Phi(z^{2})$ by $\pm\nu_{1},\ldots,\pm\nu_{l}$ such that

0<\nu_{1}<\nu_{2}<\ldots<\nu_{l}.

Note that $\nu_{1}\neq 0$ since $p_{0}(0)\neq 0$ .

So the function $z\Phi(z^{2})$ can be represented in the following form

z\Phi(z^{2})=-\alpha z+\sum_{j=1}^{l}\frac{\gamma_{j}}{z-\nu_{j}}+\sum_{j=1}^{l}\frac{\gamma_{j}}{z+\nu_{j}}=-\alpha z+\sum_{j=1}^{l}\frac{2\gamma_{j}z}{z^{2}-\nu_{j}^{2}},\quad\alpha\geqslant 0,\,\gamma_{j},\nu_{j}>0.

Divide this equality by $z$ and changing variables as follows $z^{2}\to u$ , $2\gamma_{j}\to\beta_{j}$ , $\nu_{j}^{2}\to\omega_{j}$ , we obtain the following representation of the function $\Phi$ :

\Phi(u)=-\alpha+\sum_{j=1}^{l}\frac{\beta_{j}}{u-\omega_{j}},

(4.5)

where $\alpha\geqslant 0,\,\beta_{j}>0$ and

0<\omega_{1}<\omega_{2}<\ldots<\omega_{l}.

Here $\alpha=0$ whenever $n=2l$ , and $-\alpha=\displaystyle\frac{a_{0}}{a_{1}}<0$ whenever $n=2l+1$ . Since $\Phi$ can be represented in the form (4.5), by Theorem 1.18, $\Phi$ is an R-function with exactly $l$ poles. Moreover, from (4.5) it also follows that all poles $\Phi$ are positive and $\displaystyle\lim_{u\to\pm\infty}\Phi(u)<0$ if $n=2l+1$ .

Conversely, let the polynomial $p$ be defined in (2.1), and let its associated function $\Phi$ be an R-function with exactly $l$ poles, all of which are positive, and let $\displaystyle\lim_{u\to\pm\infty}\Phi(u)<0$ when $n=2l+1$ . We will show that $p$ is self-interlacing of type I.

By Theorem 1.18, $\Phi$ can be represented in the form (4.5), where $-\alpha=\displaystyle\lim_{u\to\pm\infty}\Phi(u)\leqslant 0$ such that $\alpha>0$ if $n=2l+1$ , and $\alpha=0$ if $n=2l$ . Thus, the polynomial $p_{0}$ has only positive zeroes, and the polynomials $p_{0}$ and $p_{1}$ have no common zeroes. Together with (2.3) and (2.7), this means that the set of zeroes of the polynomial $p$ coincides with the set of roots of the equation

z\Phi(z^{2})=-1.

(4.6)

At first, we study real zeroes of this equation, so we consider only real $z$ . Since $p_{0}$ has only positive zeroes, we have $p(0)=p_{0}(0)\neq 0$ . Thus, we put $z\in\mathbb{R}\backslash\{0\}$ . Changing variables as follows $z^{2}\to u>0$ , we rewrite the equation (4.6) in the following form, which is equivalent to (4.6) for real nonzero $z$ :

\begin{cases}\ \Phi(u)=-\dfrac{1}{\sqrt{u}},\\ \ \Phi(u)=\dfrac{1}{\sqrt{u}},\end{cases}\quad u>0.

(4.7)

Note that all positive roots (if any) of the first equation (4.7) are squares of the positive roots of the equation (4.6), and all positive roots (if any) of the second equation (4.7) are squares of the negative roots of the equation (4.6).

Let $n=2l$ . Then $\lim\limits_{u\to\pm\infty}\Phi(u)=-\alpha=0$ . Consider the function $F_{1}(u)=\Phi(u)+\dfrac{1}{\sqrt{u}}$ whose set of zeroes coincides with the set of roots of the first equation (4.7). The function $F_{1}(u)$ has the same positive poles as $\Phi(u)$ does, that is, $\omega_{1},\omega_{2},\ldots,\omega_{l}$ . Moreover, $F_{1}(u)$ is decreasing on the intervals $(0,\omega_{1}),(\omega_{1},\omega_{2}),\ldots,(\omega_{l-1},\omega_{l})$ , $(\omega_{l},+\infty)$ as a sum of functions that are decreasing on those intervals. Since $F_{1}$ is decreasing on $(\omega_{l},+\infty)$ and $F_{1}(u)\to+0$ as $u\to+\infty$ , we have $F_{1}(u)>0$ for $u\in(\omega_{l},+\infty)$ . Further, $F_{1}(u)\to-\infty$ as $u\nearrow\omega_{i}$ and $F_{1}(u)\to+\infty$ as $u\searrow\omega_{i}$ , $i=1,\ldots,l$ . Also $F_{1}(u)\to+\infty$ whenever $u\to+0$ , since $|\Phi(0)|=\left|\dfrac{p_{1}(0)}{p_{0}(0)}\right|<\infty$ and $\dfrac{1}{\sqrt{u}}\to+\infty$ as $u\to+0$ . Thus, on each of the intervals $(0,\omega_{1}),(\omega_{1},\omega_{2}),\ldots,(\omega_{l-1},\omega_{l})$ , the function $F_{1}$ decreases from $+\infty$ to $-\infty$ , and therefore, it has exactly one zero, counting multiplicity, on each of those intervals. Denote the zero of $F_{1}$ on the interval $(\omega_{i-1},\omega_{i})$ by $\mu_{i}^{2}$ , $i=2,\ldots,l$ ( $\mu_{i}>0$ ). Also we denote by $\mu_{1}^{2}$ ( $\mu_{1}>0$ ) the zero of $F_{1}$ on the interval $(0,\omega_{1})$ . Since $\Phi$ is an R-function by assumption, it has exactly one zero, counting multiplicity, say $\xi_{i}$ , on each of the intervals $(\omega_{i},\omega_{i+1})$ , $i=1,\ldots,l-1$ . But $F_{1}(\xi_{i})=\dfrac{1}{\sqrt{\xi_{i}}}>0$ , so we have

\mu^{2}_{1}<\omega_{1}<\xi_{1}<\mu^{2}_{2}<\omega_{2}<\xi_{2}<\ldots<\xi_{l-1}<\mu_{l}^{2}<\omega_{l}.

(4.8)

Thus, the first equation (4.7) has exactly $l$ positive roots $\mu_{i}$ , all of which are simple and satisfy the inequalities (4.8).

Consider now the function $F_{2}(u)=\Phi(u)-\dfrac{1}{\sqrt{u}}$ whose set of zeroes coincides with the set of roots of the second equation (4.7). Since $\Phi(0)=-\displaystyle\sum_{i=1}^{l}\dfrac{\beta_{i}}{\omega_{i}}<0$ by (4.5), and $\Phi(u)\to-\infty$ as $u\nearrow\omega_{1}$ , the function $\Phi$ decreases from $\Phi(0)<0$ to $-\infty$ on the interval $(0,\omega_{1})$ . Consequently, $F_{2}(u)<0$ for all $u\in(0,\omega_{1})$ . It is clear that $F_{2}(u)\to+\infty$ as $u\searrow\omega_{i}$ and $F_{2}(u)\to-\infty$ as $u\nearrow\omega_{i}$ , $i=1,\ldots,l$ . Since $F_{2}(\mu^{2}_{i})=-\dfrac{1}{\mu_{i}}<0$ , the function $F_{2}$ has an odd number of zeroes, counting multiplicities, on each of the intervals $(\omega_{i},\mu^{2}_{i})$ , $i=1,\ldots,l-1$ . Besides, $F_{2}(u)=F_{1}(u)-\dfrac{2}{\sqrt{u}}<0$ whenever $u\in[\mu^{2}_{i},\omega_{i+1})$ , $i=1,\ldots,l-1$ , because $F_{1}(u)<0$ on those intervals. More exactly, since $F_{2}(\xi_{i})=\dfrac{1}{\sqrt{\xi_{i}}}<0$ , $i=1,\ldots,r-1$ , the function $F_{2}(u)$ has an odd number of zeroes on each interval $(\omega_{i},\xi_{i})$ and an even number of zeroes on each interval $[\xi_{i},\mu^{2}_{i+1})$ , $i=1,\ldots,r-1$ . So $F_{2}$ has at least $l-1$ zeroes in the interval $(\omega_{1},\omega_{l})$ . Consider now the interval $(\omega_{l},+\infty)$ and show that $F_{2}(u)\to-0$ as $u\to+\infty$ . In fact, since the function $\Phi$ is an R-function by assumption, we have $s_{0}=\lim\limits_{u\to\pm\infty}u\Phi(u)=D_{1}(\Phi)>0$ according Theorem 1.18, where $s_{0}$ is the first coefficient¹⁰¹⁰10Recall that $s_{-1}=0$ whenever $n=2l$ . in the series (2.13). Therefore, $\Phi(u)\sim u^{-1}$ as $u\to+\infty$ . This implies the following: $F_{2}(u)\sim-u^{-\tfrac{1}{2}}$ as $u\to+\infty$ . Thus, $F_{2}(\omega_{l}+\varepsilon)>0$ for all sufficiently small $\varepsilon>0$ and $F_{2}(u)<0$ for all sufficiently large positive $u$ . Consequently, $F_{2}$ has an odd number of zeroes, counting multiplicities, in the interval $(\omega_{l},+\infty)$ . So $F_{2}$ has at least $l$ zeroes in the interval $(\omega_{1},+\infty)$ , and it has no zeroes in $(0,\omega_{1})$ .

Since the first equation (4.7) has exactly $l$ positive roots and the second equation (4.7) has at least $l$ positive roots, the equation (4.6) has at least $2l$ real roots, all of which are the zeroes of the polynomial $p$ . But $\deg p=2l$ by assumption, therefore, the second equation (4.7) also has exactly $l$ positive roots. Moreover, it has exactly one simple root in each of the intervals $(\omega_{i},\xi_{i})$ , $i=1,\dots,l-1$ , and exactly one simple root in the interval $(\omega_{l},+\infty)$ . Thus, denoting the positive roots of the second equation (4.7) by $\zeta_{i}^{2}$ , $i=1,\ldots,l$ ( $\zeta_{i}<0$ ), we have

\omega_{1}<\zeta_{1}^{2}<\xi_{1}<\omega_{2}<\zeta_{2}^{2}<\xi_{2}<\ldots<\xi_{l-1}<\omega_{l}<\zeta_{l}^{2}.

(4.9)

Now from (4.8)–(4.9) we obtain

0<\mu_{1}<-\zeta_{1}<\mu_{2}<-\zeta_{2}\ldots<\mu_{l}<-\zeta_{l}.

(4.10)

Recall that $\mu_{i}>0$ are the positive roots of the equation (4.6), and $\zeta_{i}<0$ are the negative roots of the equation (4.6). Thus, all roots of the equation (4.6) (and therefore, all zeroes of the polynomial $p$ ) are real and simple. Denote them by $\lambda_{i}$ and enumerate such that

0<|\lambda_{1}|<|\lambda_{2}|\ldots<|\lambda_{n}|.

Then we have $\lambda_{2i-1}=\mu_{i}>0$ and $\lambda_{2i}=\zeta_{i}<0$ , $i=1,\ldots,l$ , and the inequalities (4.10) imply (4.1). Thus, $p$ is a self-interlacing polynomial of type I.

In the same way, one can show that if $n=2l+1$ , the function $\Phi$ is an R-function with positive poles, and $\lim\limits_{u\to\pm\infty}\Phi(u)<0$ , then the polynomial $p$ is self-interlacing of type I. ∎

Remark 4.4.

Let us note that if the even and odd parts, $p_{0}$ and $p_{1}$ , of a given polynomial $p$ have positive interlacing zeroes, then the polynomials $p_{0}(z^{2})$ and $zp_{1}(z^{2})$ have real interlacing zeroes, and therefore the polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ has real zeroes by Theorem 1.18 and Corollary 1.19 as a linear combination of polynomials with real interlacing zeroes. However, this notice does not help to investigate the self-interlacing property of polynomials.

Remark 4.5.

In the proof of Theorem 4.3, we also established that the squares of both positive and negative zeroes of the self-interlacing polynomial $p$ interlace both zeroes of $p_{0}$ and zeroes of $p_{1}$ .

Theorem 4.3 allow us to use properties of R-functions in order to obtain additional criteria of self-interlacing. At first, consider the Hankel matrix $S(\Phi)=\|s_{i+j}\|^{\infty}_{i,j=0}$ constructed with the coefficients of the series (2.13). From Theorems 1.18, 4.3 and 1.11 and from Corollary 1.25 we obtain the following self-interlacing criteria:

Theorem 4.6.

Let a real polynomial $p$ be defined by (2.1). The following conditions are equivalent:

1)

the polynomial $p$ is self-interlacing of type I;

the following hold

\begin{split}&s_{-1}<0\quad\text{for}\quad n=2l+1,\\ &D_{j}(\Phi)>0,\qquad j=1,\ldots,l,\\ &\widehat{D}_{j}(\Phi)>0,\qquad j=1,\ldots,l,\end{split}

(4.11)

where $l=\left[\dfrac{n}{2}\right]$ ;

3)

the matrix $S(\Phi)$ is strictly totally positive of rank $l$ , and $s_{-1}<0$ whenever $n=2l+1$ .

The determinants $D_{j}(\Phi)$ and $\widehat{D}_{j}(\Phi)$ are defined in (1.5) and (1.8), respectively.

Proof.

According to Theorem 4.3, the polynomial $p$ is self-interlacing of type I if and only if its associated function $\Phi$ is an R-function with exactly $l=\left[\dfrac{n}{2}\right]$ poles, all of which are positive, and $\lim\limits_{u\to\pm\infty}\Phi(u)=s_{-1}\leqslant 0$ . By Theorem 1.18 and Corollary 1.25, this is equivalent to the inequalities (4.11). But these inequalities are equivalent to the strictly total positivity of the matrix $S(\Phi)$ by Theorem 1.11. ∎

From this theorem we obtain the following criterion of self-interlacing, which is an analogue of the Hurwitz stability criterion.

Theorem 4.7.

Given a real polynomial $p$ of degree $n$ as in (2.1), the following conditions are equivalent:

1)

$p$ is self-interlacing of type I;

the Hurwitz determinants $\Delta_{j}(p)$ satisfy the inequalities:

\Delta_{n-1}(p)>0,\ \Delta_{n-3}(p)>0,\ldots,

(4.12)

\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]}\Delta_{n}(p)>0,\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-1}\Delta_{n-2}(p)>0,\ldots

(4.13)

the determinants $\eta_{j}(p)$ up to order $n+1$ satisfy the inequalities:

\eta_{n}(p)>0,\ \eta_{n-2}(p)>0,\ldots,\ \eta_{1}(p)=a_{0}>0,

(4.14)

\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]}\eta_{n+1}(p)>0,\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-1}\eta_{n-1}(p)>0,\ldots

(4.15)

Proof.

We establish the equivalence $1)\Longleftrightarrow 2)$ while the equivalence $2)\Longleftrightarrow 3)$ follows from (2.12).

By Theorem 4.6, $p$ is self-interlacing of type I if and only if the inequalities (4.11) hold. Now from (4.11) and (2.14)–(2.15) we obtain that $p$ is self-interlacing of type I if and only if the inequalities (4.12) hold and

\begin{array}[]{l}\text{for}\qquad n=2l\qquad\qquad\qquad(-1)^{j}\Delta_{2j}(p)>0,\qquad j=1,\ldots,l;\\ \\ \text{for}\qquad n=2l+1\qquad\qquad\;(-1)^{j+1}\Delta_{2j+1}(p)>0,\qquad j=0,1,\ldots,l,\end{array}

that is equivalent to (4.13). ∎

Note that the inequalities (4.13) and (4.15) are equivalent to the following ones:

\Delta_{n-2i}(p)\Delta_{n-2i-2}(p)<0,\quad\text{and}\quad\eta_{n-2i+1}(p)\eta_{n-2i-1}(p)<0,\quad i=0,1,\ldots,\left[\dfrac{n-1}{2}\right],

where $\Delta_{0}(p)=\eta_{0}(p)=a_{0}\Delta_{-1}(p)\equiv 1$ .

Analogues of Theorems 3.7–3.9 will be established in Section 4.3 using a connection between Hurwitz stable polynomials and self-interlacing polynomials of type I.

4.2 Interrelation between Hurwitz stable and self-interlacing polynomials

Comparing Theorems 3.3 and 4.3, one obtains the following fact, which has deep consequences and allows us to describe a lot of properties of self-interlacing polynomials.

Theorem 4.8.

A polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ is self-interlacing of type I if and only if the polynomial $q(z)=p(-z^{2})-zp_{1}(z^{2})$ is Hurwitz stable.

Proof.

By Theorem 4.3, $p$ is self-interlacing if and only if the function $\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}$ is an R-function with only positive poles, and $\lim\limits_{u\to\pm\infty}\Phi(u)\leqslant 0$ that is equivalent to the fact that the function $\Psi(u)=-\dfrac{p_{1}(-u)}{p_{0}(-u)}$ is an R-function with only negative poles, and $\lim\limits_{u\to\pm\infty}\Psi(u)\geqslant 0$ . According to Theorem 3.3, this is equivalent to the polynomial $q$ being Hurwitz stable, as required. ∎

Remark 4.9.

Thus, we have that there is one-to-one correspondence between self-interlacing and Hurwitz stable polynomials. Given a real Hurwitz stable polynomials, we should change sings of some its coefficients to obtain a self-interlacing polynomial, and vise versa.

As an immediate consequence of Theorem 4.8, we obtain the following analogue of Stodola’s theorem, Theorem 3.2.

Theorem 4.10.

If the polynomial

p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0,

(4.16)

is self-interlacing of type I, then

for $n=2l$ ,

$(-1)^{\tfrac{j(j-1)}{2}}a_{j}>0,\qquad j=0,1,\ldots,n;$
for $n=2l+1$ ,

$(-1)^{\tfrac{j(j+1)}{2}}a_{j}>0,\qquad j=0,1,\ldots,n.$

Proof.

Since $p$ is self-interlacing of type I, by Theorem 4.8, the polynomial¹¹¹¹11We put the factor $(-1)^{\tfrac{n(n+1)}{2}}$ in order to make the leading coefficient of the polynomial $q$ equal to $a_{0}>0$ .

q(z)=(-1)^{\tfrac{n(n+1)}{2}}[p_{0}(-z^{2})-zp_{1}(-z^{2})]=b_{0}z^{n}+b_{1}z^{n-1}+b_{2}z^{n-2}+\dots+b_{n},\quad b_{0}=a_{0}>0,

(4.17)

is Hurwitz stable. It is easy to see that

b_{j}=(-1)^{\tfrac{j(j-1)}{2}}a_{j},\qquad j=0,1,\ldots,n\qquad\qquad\qquad\quad\quad\text{for}\qquad n=2l;

(4.18)

and

\quad\,\,\,\,\qquad\qquad\qquad b_{j}=(-1)^{\tfrac{j(j+1)}{2}}a_{j},\qquad j=0,1,\ldots,n\qquad\qquad\qquad\qquad\text{for}\qquad n=2l+1.

(4.19)

By Stodola’s theorem, Theorem 3.2, all $b_{j}$ are positive. ∎

Thus, a necessary form of a self-interlacing polynomial with positive leading coefficient is as follows:
for $n=2l$

p(z)=b_{0}z^{n}+b_{1}z^{n-1}-b_{2}z^{n-2}-b_{3}z^{n-3}+b_{4}z^{n-4}+b_{5}z^{n-5}-b_{6}z^{n-6}-\dots,\qquad b_{0},\dots,b_{n}>0,

for $n=2l+1$

p(z)=b_{0}z^{n}-b_{1}z^{n-1}-b_{2}z^{n-2}+b_{3}z^{n-3}+b_{4}z^{n-4}-b_{5}z^{n-5}-b_{6}z^{n-6}+\dots,\qquad b_{0},\dots,b_{n}>0.

We also notice that if the polynomial $p$ defined in (4.16) is self-interlacing, then

a_{n-1}a_{n}<0,

(4.20)

since $\Psi(0)=-\Phi(0)=-{p_{1}(0)}/{p_{0}(0)}=-{a_{n-1}}/{a_{n}}>0$ (see the proof of Theorem 4.10).

Remark 4.11.

In fact, Theorem 4.10 was proved in [8] by another methods.

Let us point out at one more interesting connection between Hurwitz stable and self-interlacing polynomials. Let the polynomial $p$ be self-interlacing. If $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ , then $p(iz)=p_{0}(-z^{2})+izp_{1}(-z^{2})$ , where $i=\sqrt{-1}$ . Consequently, the Hurwitz stable polynomials $q(z)=p_{0}(-z^{2})-zp_{1}(-z^{2})$ can be represented as follows

q(z)=\dfrac{p(iz)+p(-iz)}{2}-\dfrac{p(iz)-p(-iz)}{2i}=p(iz)\dfrac{1+i}{2}+p(-iz)\dfrac{1-i}{2}.

So one can establish the following theorem.

Theorem 4.12.

Let the polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ be self-interlacing of type I and let its dual polynomial $q(z)=p_{0}(-z^{2})-zp_{1}(-z^{2})$ be the Hurwitz stable polynomial associated with $p$ . Then

p(\lambda)=0\Longleftrightarrow\arg q(i\lambda)=\dfrac{\pi}{4}\,\,\,\text{or}\,\,\,\dfrac{5\pi}{4};

and, respectively,

q(\mu)=0\Longleftrightarrow\arg p(i\mu)=\dfrac{\pi}{4}\,\,\,\text{or}\,\,\,\dfrac{5\pi}{4}.

Proof.

In fact, $p(\lambda)=0$ if and only if $-\dfrac{\lambda p_{1}(\lambda^{2})}{p_{0}(\lambda^{2})}=1$ . At the same time, $q(i\lambda)=p_{0}(\lambda^{2})-i\lambda p_{1}(\lambda^{2})$ . Consequently, $\arg q(i\lambda)=\arctan\left(-\dfrac{\lambda p_{1}(\lambda^{2})}{p_{0}(\lambda^{2})}\right)=\arctan 1=\dfrac{\pi}{4}\,\text{or}\,\dfrac{5\pi}{4}$ .

The second assertion of the theorem can be proved analogously. ∎

4.3 Liénard and Chipart criterion, Stieltjes continued fractions and the signs of Hurwitz minors

Now we consider the polynomials $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ and $q(z)=p_{0}(-z^{2})-zp_{1}(-z^{2})$ and their associated functions $\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}$ and $\Psi(u)=-\dfrac{p_{1}(-u)}{p_{0}(-u)}$ . Let

\Phi(u)=s_{-1}+\frac{s_{0}}{u}+\frac{s_{1}}{u^{2}}+\frac{s_{2}}{u^{3}}+\frac{s_{3}}{u^{4}}+\dots

Then we have

\Psi(u)=-\dfrac{p_{1}(-u)}{p_{0}(-u)}=t_{-1}+\frac{t_{0}}{w}+\frac{t_{1}}{w^{2}}+\frac{t_{2}}{w^{3}}+\frac{t_{3}}{w^{4}}+\frac{t_{3}}{w^{5}}+\dots=-s_{-1}+\frac{s_{0}}{w}-\frac{s_{1}}{w^{2}}+\frac{s_{2}}{w^{3}}-\frac{s_{3}}{w^{4}}+\frac{s_{3}}{w^{5}}-\dots

Thus, the connection between $s_{j}$ and $t_{j}$ is as follows

t_{j}=(-1)^{j}s_{j},\qquad j=-1,0,1,2,\ldots

Consider the two infinite Hankel matrices $S=\|s_{i+j}\|_{0}^{\infty}$ and $T=\|t_{i+j}\|_{0}^{\infty}$ .

In [16] there was proved the following formula.

Theorem 4.13.

Minors of the matrices $S$ and $T$ are connected as follows

T\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\\ \end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}i_{k}+\sum\limits_{k=1}^{m}j_{k}}S\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\\ \end{pmatrix},\quad m=1,2,\ldots

From this theorem one can easily obtain the following consequence.

Corollary 4.14.

Let rational functions $\Phi(u)$ and $\Psi(u)$ be such that $\Psi(u)=-\Phi(-u)$ . Then the following equalities hold

D_{j}(\Phi)=D_{j}(\Psi),\quad j=1,2,\ldots

\widehat{D}_{j}(\Phi)=(-1)^{j}\widehat{D}_{j}(\Psi),\quad j=1,2,\ldots

From this corollary and from (2.14)–(2.15) one obtains

\Delta_{n+1-2j}(q)=\Delta_{n+1-2j}(p),\qquad j=1,\ldots,\left[\dfrac{n}{2}\right],

(4.21)

\Delta_{n-2j}(q)=\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-j}\Delta_{n-2j}(p),\qquad j=0,1,\ldots,\left[\dfrac{n}{2}\right],

(4.22)

where the polynomials $p$ and $q$ are defined in (4.16) and (4.17), respectively.

Now we are in position to prove analogues of Theorems 3.7–3.9 for self-interlacing polynomials.

Theorem 4.15.

The polynomial $p$ defined in (4.16) is self-interlacing of type I if and only if

(-1)^{\left[\tfrac{n+1}{2}\right]}a_{n}>0,\,(-1)^{\left[\tfrac{n+1}{2}\right]-1}a_{n-2}>0,\,(-1)^{\left[\tfrac{n+1}{2}\right]-2}a_{n-4}>0,\,\dots

(4.23)

(-1)^{\left[\tfrac{n+1}{2}\right]}a_{n}>0,\,(-1)^{\left[\tfrac{n+1}{2}\right]-1}a_{n-1}>0,\,(-1)^{\left[\tfrac{n+1}{2}\right]-2}a_{n-3}>0,\,\dots

(4.24)

and one of the following two conditions holds

1)

$\Delta_{n-1}(p)>0,\,\Delta_{n-3}(p)>0,\,\Delta_{n-5}(p)>0,\,\dots;$ (4.25)

\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]}\Delta_{n}(p)>0,\,\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-1}\Delta_{n-2}(p)>0,\,\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-2}\Delta_{n-4}(p)>0,\,\ldots

(4.26)

Proof.

If the polynomial $p$ is self-interlacing of type I, then by Theorems 4.7 and 4.10, all the conditions (4.23)–(4.26) hold.

Let the conditions (4.23) and (4.25) hold. Consider the polynomial $q$ defined in (4.17). From (4.23) and (4.18)–(4.19) it follows that $b_{n}>0,b_{n-2}>0,b_{n-4}>0,\ldots$ , and from (4.25), (2.14)–(2.15) and (4.21) we obtain that $\Delta_{n-1}(q)>0,\Delta_{n-3}(q)>0,\ldots$ Thus, the polynomial $q$ satisfies the condition $1)$ of Theorem 3.7. Therefore, $q$ is Hurwitz stable, so $p$ is self-interlacing of type I according to Theorem 4.8.

Analogously, using (4.18)–(4.19), (2.14)–(2.15), (4.21)–(4.22) and Theorem 3.7 one can show that the conditions (4.24) and (4.25), or (4.23) and (4.26), or (4.24) and (4.26) imply Hurwitz stability of the polynomial $q$ , so by Theorem 4.8, $p$ is self-interlacing of type I. ∎

The following theorem presents a relation between self-interlacing polynomials and continued fractions of Stieltjes type.

Theorem 4.16.

The polynomial $p$ of degree $n$ defined in (4.16) is self-interlacing of type I if and only if its associated function $\Phi$ has the following Stieltjes continued fraction expansion:

\Phi(u)=c_{0}+\dfrac{1}{c_{1}u+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}u+\cfrac{1}{\ddots+\cfrac{1}{c_{2l-1}u+\cfrac{1}{c_{2l}}}}}}},\quad\text{with}\quad(-1)^{i-1}c_{i}>0,\quad i=1,\ldots,2l,

(4.27)

where $c_{0}=0$ if $n$ is even, and $c_{0}>0$ if $n$ is odd, and $l$ as in (2.2).

Proof.

In fact, by Theorem 4.3, the polynomial $p$ is self-interlacing of type I if and only if the function $\Phi$ can be represented in the form (4.5), where $-\alpha=c_{0}=0$ if $n=2l$ , and $-\alpha=c_{0}=\dfrac{a_{0}}{a_{1}}<0$ if $n=2l+1$ . Now the assertion of the theorem follows from Theorem 1.18 and Corollary 1.39. ∎

Theorems 3.9 and 4.8 imply the following theorem.

Theorem 4.17.

Let $p$ be a self-interlacing polynomial of type I of degree $n\geqslant 2$ as in (4.16). Then all the polynomials

p_{j}(z)=\sum\limits_{i=0}^{n-2j}\left[\dfrac{n-i}{2}\right]\left(\left[\dfrac{n-i}{2}\right]-1\right)\cdots\left(\left[\dfrac{n-i}{2}\right]+j-1\right)a_{i}z^{n-2j-i},\quad j=1,\ldots,\left[\dfrac{n}{2}\right]-1,

also are self-interlacing of type I (if $j$ is even) or of type II (if $j$ is odd).

Let again $p$ be a self-interlacing polynomial of type Iof degree $n$ and let $q$ be its associated Hurwitz stable polynomial, that is, $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ and $q(z)=(-1)^{\tfrac{n(n+1)}{2}}[p_{0}(-z^{2})-zp_{1}(-z^{2})]$ . We are in a position to find an interrelation between Hurwitz minors of these polynomials.

Let $n=2l$ . By Theorem 4.8, the Hurwitz matrix of the polynomial $q$ has the form

\mathcal{H}_{n}(q)=\begin{pmatrix}a_{1}&-a_{3}&a_{5}&-a_{7}&\dots&0\\ a_{0}&-a_{2}&a_{4}&-a_{6}&\dots&0\\ 0&a_{1}&-a_{3}&a_{5}&\dots&0\\ 0&a_{0}&-a_{2}&a_{4}&\dots&0\\ \dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&(-1)^{\left[\frac{n}{2}\right]}a_{n}\\ \end{pmatrix}.

It is easy to see that the matrix $\mathcal{H}_{n}(q)$ can be factorized as follows

\mathcal{H}_{n}(q)=\widetilde{C}_{n}\mathcal{H}_{n}(p)\widetilde{E}_{n},

(4.28)

where the $n\times n$ matrices $\widetilde{E}_{n}$ and $\widetilde{C}_{n}$ have the forms

\widetilde{E}_{n}=\begin{pmatrix}1&0&0&0&0&\dots\\ 0&-1&0&0&0&\dots\\ 0&0&1&0&0&\dots\\ 0&0&0&-1&0&\dots\\ 0&0&0&0&1&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix},\qquad\widetilde{C}_{n}=\begin{pmatrix}1&0&0&0&0&\dots\\ 0&1&0&0&0&\dots\\ 0&0&-1&0&0&\dots\\ 0&0&0&-1&0&\dots\\ 0&0&0&0&1&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.

All non-principal minors of these matrices are equal to zero. So since $\widetilde{e}_{jj}=(-1)^{j-1}$ and $\widetilde{c}_{jj}=(-1)^{\tfrac{(j-1)(j-2)}{2}}$ , we have

\widetilde{C}_{n}\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ i_{1}&i_{2}&\dots&i_{m}\end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}\tfrac{(i_{k}-1)(i_{k}-2)}{2}},\qquad\widetilde{E}_{n}\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ i_{1}&i_{2}&\dots&i_{m}\end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}i_{k}-m},

(4.29)

where $1\leqslant i_{1}<i_{2}<\ldots<i_{m}\leqslant n$ . Thus, the Cauchy–Binet formula together with (4.29) and (4.28) implies, for $n=2l$ ,

\mathcal{H}_{n}(q)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}\tfrac{(i_{k}-1)(i_{k}-2)}{2}+\sum\limits_{k=1}^{m}j_{k}-m}\mathcal{H}_{n}(p)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix}.

(4.30)

where $1\leqslant\begin{array}[]{c}i_{1}<i_{2}<\ldots<i_{m}\\ j_{1}<j_{2}<\ldots<j_{m}\end{array}\leqslant n$ ,

Let now $n=2l+1$ . Then by Theorem 4.8, the Hurwitz matrix of the polynomial $q$ has the form

\mathcal{H}_{n}(q)=\begin{pmatrix}-a_{1}&a_{3}&-a_{5}&a_{7}&\dots&0\\ a_{0}&-a_{2}&a_{4}&-a_{6}&\dots&0\\ 0&-a_{1}&a_{3}&-a_{5}&\dots&0\\ 0&a_{0}&-a_{2}&a_{4}&\dots&0\\ \dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&(-1)^{\left[\frac{n+1}{2}\right]}a_{n}\\ \end{pmatrix}.

In this case, $\mathcal{H}_{n}(q)$ also can be factorized:

\mathcal{H}_{n}(q)=\widehat{C}_{n}\mathcal{H}_{n}(p)(-\widetilde{E}_{n}),

(4.31)

where the $n\times n$ matrix $\widehat{C}_{n}$ is as follows

\widehat{C}_{n}=\begin{pmatrix}1&0&0&0&0&0&\dots\\ 0&-1&0&0&0&0&\dots\\ 0&0&-1&0&0&0&\dots\\ 0&0&0&1&0&0&\dots\\ 0&0&0&0&1&0&\dots\\ 0&0&0&0&0&-1&\dots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}.

All non-principal minors of the matrices $\widehat{C}_{n}$ and $-\widetilde{E}_{n}$ equal zero. The principal minors of these matrices can be easily calculated:

\widehat{C}_{n}\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ i_{1}&i_{2}&\dots&i_{m}\end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}\tfrac{i_{k}(i_{k}-1)}{2}},\qquad\widehat{E}_{n}\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ i_{1}&i_{2}&\dots&i_{m}\\ \end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}i_{k}},

(4.32)

where $1\leqslant i_{1}<i_{2}<\ldots<i_{m}\leqslant n$ . Thus, the Cauchy–Binet formula together with (4.32) and (4.31) implies, for $n=2l+1$ ,

\mathcal{H}_{n}(q)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix}=(-1)^{\sum\limits_{k=1}^{m}\tfrac{i_{k}(i_{k}-1)}{2}+\sum\limits_{k=1}^{m}j_{k}}\mathcal{H}_{n}(p)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix},

(4.33)

where $1\leqslant\begin{array}[]{c}i_{1}<i_{2}<\ldots<i_{m}\\ j_{1}<j_{2}<\ldots<j_{m}\end{array}\leqslant n$ .

Since $q$ is Hurwitz stable, from Theorem 3.5 and from the formulæ (4.30) and (4.33) we obtain the following theorem.

Theorem 4.18.

Let $p$ be a self-interlacing polynomial of type I of degree $n$ and let $\mathcal{H}_{n}(p)$ be its Hurwitz matrix defined in (2.10). Then

for $n=2l$ ,

(-1)^{\sum\limits_{k=1}^{m}\tfrac{(i_{k}-1)(i_{k}-2)}{2}+\sum\limits_{k=1}^{m}j_{k}-m}\mathcal{H}_{n}(p)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix}\geqslant 0;

for $n=2l+1$ ,

(-1)^{\sum\limits_{k=1}^{m}\tfrac{i_{k}(i_{k}-1)}{2}+\sum\limits_{k=1}^{m}j_{k}}\mathcal{H}_{n}(p)\begin{pmatrix}i_{1}&i_{2}&\dots&i_{m}\\ j_{1}&j_{2}&\dots&j_{m}\end{pmatrix}\geqslant 0,

where $1\leqslant\begin{array}[]{c}i_{1}<i_{2}<\ldots<i_{m}\\ j_{1}<j_{2}<\ldots<j_{m}\end{array}\leqslant n$ .

4.4 The second proof of the Hurwitz self-interlacing criterion

Given a polynomial

p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0,

(4.34)

we consider the following rational functions

R(z)=\dfrac{(-1)^{n}p(-z)}{p(z)}\quad\text{and}\quad F(z)=\dfrac{1}{R(z)}=\dfrac{p(z)}{(-1)^{n}p(-z)}

(4.35)

and expand them into their Laurent series at $\infty$ :

R(z)=1+\frac{s_{0}}{z}+\frac{s_{1}}{z^{2}}+\frac{s_{2}}{z^{3}}+\dots,\qquad F(z)=1+\frac{t_{0}}{z}+\frac{t_{1}}{z^{2}}+\frac{t_{2}}{z^{3}}+\dots

As we mentioned in Section 1.1, ranks of the matrices $S=\|s_{j+k}\|_{0}^{\infty}$ and $T=\|t_{j+k}\|_{0}^{\infty}$ are equal to the number of poles of the functions $R$ and $F$ , respectively. It is easy to see that rank of each matrix equals $n$ if the polynomials $p(z)$ and $p(-z)$ have no common zeroes. In the rest of this section we consider only such polynomials, so in this section, ranks of the matrices $S$ and $T$ always equal $n$ . Denoting

q(z)=(-1)^{n}p(-z),

we obtain $R=\dfrac{q}{p}$ and $F=\dfrac{p}{q}$ .

Lemma 4.19.

For the functions $R$ and $F$ defined in (4.35), the following formulæ are valid:

\nabla_{2j}(p,q)=a_{0}^{2j}D_{j}(R)=(-1)^{\frac{j(j+1)}{2}}2^{j}a_{0}\Delta_{j-1}(p)\Delta_{j}(p),\quad j=1,2,\ldots

(4.36)

and

\nabla_{2j}(q,p)=a_{0}^{2j}D_{j}(F)=(-1)^{\frac{j(j-1)}{2}}2^{j}a_{0}\Delta_{j-1}(p)\Delta_{j}(p),\quad j=1,2,\ldots,

(4.37)

where the determinants $\nabla_{2j}(p,q)$ are the leading principal minors of the matrix $\mathcal{H}_{2n+1}(p,q)$ of order $2j$ , and $\Delta_{j}(p)$ are defined in (2.11), $\Delta_{0}(p)\equiv 1$ .

Proof.

We prove the formulæ (4.36). The formulæ (4.37) follow from (4.36) since $\nabla_{2j}(q,p)=(-1)^{j}\nabla_{2j}(p,q)$ .

Assuming $a_{j}=0$ for $j>n$ , we have

\nabla_{2j}(p,q)=\begin{vmatrix}a_{0}&a_{1}&a_{2}&a_{3}&\dots&a_{j-1}&a_{j}&\dots&a_{2j-2}&a_{2j-1}\\ a_{0}&-a_{1}&a_{2}&-a_{3}&\dots&-a_{j-1}&a_{j}&\dots&a_{2j-2}&-a_{2j-1}\\ 0&a_{0}&a_{1}&a_{2}&\dots&a_{j-2}&a_{j-1}&\dots&a_{2j-3}&a_{2j-2}\\ 0&a_{0}&-a_{1}&a_{2}&\dots&a_{j-2}&-a_{j-1}&\dots&-a_{2j-3}&a_{2j-2}\\ 0&0&a_{0}&a_{1}&\dots&a_{j-3}&a_{j-2}&\dots&a_{2j-4}&a_{2j-3}\\ 0&0&a_{0}&-a_{1}&\dots&-a_{j-3}&a_{j-2}&\dots&a_{2j-4}&-a_{2j-3}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&a_{0}&a_{1}&\dots&a_{j-1}&a_{j}\\ 0&0&0&0&\dots&a_{0}&-a_{1}&\dots&-a_{j-1}&a_{j}\\ \end{vmatrix}=

=2^{j}\begin{vmatrix}a_{0}&a_{1}&a_{2}&a_{3}&\dots&a_{j-1}&a_{j}&\dots&a_{2j-2}&a_{2j-1}\\ a_{0}&0&a_{2}&0&\dots&0&a_{j}&\dots&a_{2j-2}&0\\ 0&a_{0}&a_{1}&a_{2}&\dots&a_{j-2}&a_{j-1}&\dots&a_{2j-3}&a_{2j-2}\\ 0&a_{0}&0&0&\dots&a_{j-2}&0&\dots&0&a_{2j-2}\\ 0&0&a_{0}&a_{1}&\dots&a_{j-3}&a_{j-2}&\dots&a_{2j-4}&a_{2j-3}\\ 0&0&a_{0}&0&\dots&0&a_{j-2}&\dots&a_{2j-4}&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&a_{0}&a_{1}&\dots&a_{j-1}&a_{j}\\ 0&0&0&0&\dots&a_{0}&0&\dots&0&a_{j}\\ \end{vmatrix}=

=(-2)^{j}\begin{vmatrix}a_{0}&0&a_{2}&0&\dots&0&a_{j}&\dots&a_{2j-2}&0\\ 0&a_{1}&0&a_{3}&\dots&a_{j-1}&0&\dots&0&a_{2j-1}\\ 0&a_{0}&0&a_{2}&\dots&a_{j-2}&0&\dots&0&a_{2j-2}\\ 0&0&a_{1}&0&\dots&0&a_{j-1}&\dots&a_{2j-3}&0\\ 0&0&a_{0}&0&\dots&0&a_{j-2}&\dots&a_{2j-4}&0\\ 0&0&0&a_{1}&\dots&a_{j-3}&0&\dots&0&a_{2j-3}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&a_{0}&0&\dots&0&a_{j}\\ 0&0&0&0&\dots&0&a_{1}&\dots&a_{j-1}&0\\ \end{vmatrix}=

=(-2)^{j}a_{0}\begin{vmatrix}a_{1}&0&a_{3}&0&\dots&a_{j-1}&0&\dots&0&a_{2j-1}\\ a_{0}&0&a_{2}&0&\dots&a_{j-2}&0&\dots&0&a_{2j-2}\\ 0&0&a_{1}&0&\dots&a_{j-3}&0&\dots&0&a_{2j-3}\\ 0&0&a_{0}&0&\dots&a_{j-4}&0&\dots&0&a_{2j-4}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&a_{1}&0&\dots&0&a_{j+1}\\ 0&0&0&0&\dots&a_{0}&0&\dots&0&a_{j}\\ 0&a_{1}&0&a_{3}&\dots&0&a_{j-1}&\dots&a_{2j-3}&0\\ 0&a_{0}&0&a_{2}&\dots&0&a_{j-2}&\dots&a_{2j-4}&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&0&a_{2}&\dots&a_{j}&0\\ 0&0&0&0&\dots&0&a_{1}&\dots&a_{j-1}&0\\ \end{vmatrix}=

=(-2)^{j}a_{0}(-1)^{\frac{j(j-1)}{2}}\begin{vmatrix}a_{1}&a_{3}&a_{5}&\dots&a_{2j-1}&0&0&\dots&0&0\\ a_{0}&a_{2}&a_{4}&\dots&a_{2j-2}&0&0&\dots&0&0\\ 0&a_{1}&a_{3}&\dots&a_{2j-3}&0&0&\dots&0&0\\ 0&a_{0}&a_{2}&\dots&a_{2j-4}&0&0&\dots&0&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&\dots&a_{j+1}&0&0&\dots&0&0\\ 0&0&0&\dots&a_{j}&0&0&\dots&0&0\\ 0&0&0&\dots&0&a_{1}&a_{3}&\dots&a_{2j-5}&a_{2j-3}\\ 0&0&0&\dots&0&a_{0}&a_{2}&\dots&a_{2j-6}&a_{2j-4}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&\dots&0&0&0&\dots&a_{j-2}&a_{j}\\ 0&0&0&\dots&0&0&0&\dots&a_{j-3}&a_{j-1}\\ \end{vmatrix}=

=(-1)^{\frac{j(j+1)}{2}}2^{j}a_{0}\Delta_{j-1}(p)\Delta_{j}(p).

∎

Using this lemma it is easy to prove the equivalence of the conditions $1)$ and $2)$ of Theorem 4.7.

Theorem 4.20.

The polynomial $p$ defined in (4.34) is self-interlacing of type I if and only if

\Delta_{n-1}(p)>0,\ \Delta_{n-3}(p)>0,\dots,

(4.38)

\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]}\Delta_{n}(p)>0,\displaystyle(-1)^{\left[\tfrac{n+1}{2}\right]-1}\Delta_{n-2}(p)>0,\ldots

(4.39)

where the Hurwitz minors $\Delta_{i}(p)$ are defined in (2.11).

Proof.

In the proof of Theorem 4.3, it was established that if $p$ is self-interlacing of type I, then the function $-\dfrac{p(-z)}{p(z)}$ is an R-function with exactly $n$ poles.

Let $n=2l+1$ . Then the function $R$ defined in (4.35) is an R-function. By Theorem 1.18, we have $D_{j}(R)>0$ for $j=1,\ldots,n$ . These inequalities together with the formulæ (4.36) imply

(-1)^{\frac{j(j+1)}{2}}\Delta_{j-1}(p)\Delta_{j}(p)>0,\qquad j=1,\ldots,n.

(4.40)

Multiplying the inequalities (4.40) for $j=2m$ and $j=2m-1$ , we obtain

\Delta_{2m-1}^{2}\Delta_{2m}\Delta_{2m-2}>0.

Consequently, the minors $\Delta_{2i}(p)$ , $i=1,\ldots,l$ , are positive, so the inequalities (4.38) are proved for odd $n$ .

If we multiply the inequalities (4.40) for $j=2m$ and $j=2m+1$ , we get

-\Delta_{2m}^{2}(p)\Delta_{2m-1}(p)\Delta_{2m+1}(p)>0.

These inequalities imply (4.39) for odd $n$ .

The converse assertion can be proved in the same way. That is, the inequalities (4.38)–(4.39) imply the inequalities (4.40) which in turn imply the inequalities $D_{j}(R)>0$ , $j=1,\ldots,n$ , according to (4.36). By Theorem 1.18, the function $-\dfrac{p(-z)}{p(z)}$ is an R-function, so $p$ is a self-interlacing polynomial of type I (see the proof of Theorem 4.3).

If $n$ is even, the inequalities (4.38)–(4.39) can be proved analogously using the function $F$ defined in (4.35) instead of the function $R$ . ∎

From Lemma 4.19 and from Hurwitz’s stability criterion (see Theorem 3.5) it follows one more stability criterion.

Theorem 4.21.

1)

The polynomial $p$ is Hurwitz stable if and only if for the function $R$ defined in (4.35), the following inequalities hold

$(-1)^{\frac{j(j+1)}{2}}D_{j}(R)>0,\quad j=1,2,\ldots,n.$
2)

The polynomial $p$ is Hurwitz stable if and only if for the function $F$ defined in (4.35), the following inequalities hold

$(-1)^{\frac{j(j-1)}{2}}D_{j}(F)>0,\quad j=1,2,\ldots,n.$

Finally, we establish a relationships between the Hurwitz minors $\Delta_{j}(p)$ of a given polynomial $p$ and the Hankel minors $\widehat{D}_{j}(R)$ and $\widehat{D}_{j}(F)$ (for the definition of the minors $\widehat{D}_{j}$ see (1.8)). To do this, consider the functions

zR(z)=z+\dfrac{h_{1}(z)}{p(z)}=z+\dfrac{-2a_{1}z^{n}-2a_{3}z^{n-2}-2a_{5}z^{n-4}-\dots}{a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n}}=z+s_{0}+\frac{s_{1}}{z}+\frac{s_{2}}{z^{2}}+\frac{s_{3}}{z^{3}}+\dots

(4.41)

zF(z)=z+\dfrac{h_{2}(z)}{(-1)^{n}p(-z)}=z+\dfrac{2a_{1}z^{n}+2a_{3}z^{n-2}+2a_{5}z^{n-4}+\dots}{a_{0}z^{n}-a_{1}z^{n-1}+\dots+(-1)^{n}a_{n}}=z+t_{0}+\frac{t_{1}}{z}+\frac{t_{2}}{z^{2}}+\frac{t_{3}}{z^{3}}+\dots

(4.42)

where $s_{0}=t_{0}=1$ , and note that $\widehat{D}_{j}(R)=D_{j}(zR)$ and $\widehat{D}_{j}(F)=D_{j}(zF)$ . This allows us to establish the following lemma.

Lemma 4.22.

Given the polynomial $p$ , for the functions $R$ and $F$ defined in (4.35), the following relationships hold

\nabla_{2j}(p,h_{1})=a_{0}^{2j}\widehat{D}_{j}(R)=(-1)^{\frac{j(j-1)}{2}}2^{j}\Delta_{j}^{2}(p),\quad j=1,2,\ldots,

(4.43)

and

\nabla_{2j}(p,h_{2})=a_{0}^{2j}\widehat{D}_{j}(F)=(-1)^{\frac{j(j-1)}{2}}2^{j}\Delta_{j}^{2}(p),\quad j=1,2,\ldots,

(4.44)

where the determinants¹²¹²12The polynomials $h_{1}$ and $h_{2}$ are defined in (4.41) and (4.42), respectively. $\nabla_{2j}(p,h_{1})$ and $\nabla_{2j}(p,h_{2})$ are the leading principal minors of order $2j$ of the matrices $\mathcal{H}_{2n+1}(p,h_{1})$ and $\mathcal{H}_{2n+1}(p,h_{2})$ respectively, and $\Delta_{j}(p)$ are the Hurwitz minors of the polynomial $p$ , $\Delta_{0}(p)\equiv 1$ .

Proof.

Assuming $a_{j}=0$ for $j>n$ , from (4.41) we obtain

\nabla_{2j}(p,h_{1})=(-2)^{j}\begin{vmatrix}a_{0}&a_{1}&a_{2}&a_{3}&\dots&a_{j-1}&a_{j}&\dots&a_{2j-2}&a_{2j-1}\\ a_{1}&0&a_{3}&0&\dots&0&a_{j+1}&\dots&a_{2j-1}&0\\ 0&a_{0}&a_{1}&a_{2}&\dots&a_{j-2}&a_{j-1}&\dots&a_{2j-3}&a_{2j-2}\\ 0&a_{1}&0&a_{3}&\dots&a_{j-1}&0&\dots&0&a_{2j-1}\\ 0&0&a_{0}&a_{1}&\dots&a_{j-3}&a_{j-2}&\dots&a_{2j-4}&a_{2j-3}\\ 0&0&a_{1}&0&\dots&0&a_{j-1}&\dots&a_{2j-3}&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&a_{0}&a_{1}&\dots&a_{j-1}&a_{j}\\ 0&0&0&0&\dots&a_{1}&0&\dots&0&a_{j+1}\\ \end{vmatrix}=

=2^{j}\begin{vmatrix}a_{1}&0&a_{3}&0&\dots&0&a_{j+1}&\dots&a_{2j-1}&0\\ a_{0}&0&a_{2}&0&\dots&0&a_{j}&\dots&a_{2j-2}&0\\ 0&a_{1}&0&a_{3}&\dots&a_{j-1}&0&\dots&0&a_{2j-1}\\ 0&a_{0}&0&a_{2}&\dots&a_{j-2}&0&\dots&0&a_{2j-2}\\ 0&0&a_{1}&0&\dots&0&a_{j-1}&\dots&a_{2j-3}&0\\ 0&0&a_{0}&0&\dots&0&a_{j-2}&\dots&a_{2j-4}&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&0&\dots&0&a_{2}&\dots&a_{j}&0\\ 0&0&0&0&\dots&a_{1}&0&\dots&0&a_{j+1}\\ 0&0&0&0&\dots&a_{0}&a_{1}&\dots&a_{j-1}&a_{j}\\ \end{vmatrix}=

=2^{j}(-1)^{\frac{j(j-1)}{2}}\begin{vmatrix}a_{1}&a_{3}&a_{5}&\dots&a_{2j-1}&0&0&\dots&0&0\\ a_{0}&a_{2}&a_{4}&\dots&a_{2j-2}&0&0&\dots&0&0\\ 0&a_{1}&a_{3}&\dots&a_{2j-3}&0&0&\dots&0&0\\ 0&a_{0}&a_{2}&\dots&a_{2j-4}&0&0&\dots&0&0\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&\dots&a_{j+1}&0&0&\dots&0&0\\ 0&0&0&\dots&a_{j}&0&0&\dots&0&0\\ 0&0&0&\dots&0&a_{1}&a_{3}&\dots&a_{2j-3}&a_{2j-1}\\ 0&0&0&\dots&0&a_{0}&a_{2}&\dots&a_{2j-4}&a_{2j-2}\\ \dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\ 0&0&0&\dots&0&0&0&\dots&a_{j-1}&a_{j+1}\\ 0&0&0&\dots&a_{j-1}&0&0&\dots&a_{j-2}&a_{j}\\ \end{vmatrix}=

=(-1)^{\frac{j(j-1)}{2}}2^{j}\Delta_{j}^{2}(p).

The relationships (4.44) can be proved analogously. ∎

Thus, we obtain that for any real polynomial $p$ of degree $n$ with nonzero Hurwitz minors and, in particular, for Hurwitz stable and for self-interlacing polynomials, the following inequalities hold

(-1)^{\frac{j(j-1)}{2}}\widehat{D}_{j}(R)=(-1)^{\frac{j(j-1)}{2}}\widehat{D}_{j}(F)>0,\quad j=1,2,\ldots,n.

Last, let us note that the formulæ (4.36)–(4.37) and (4.43)–(4.44) can be obtained (overcoming certain difficulties) from some theorems of the book [41]. But it was more simple to deduce them directly as we did in Lemmata 4.19 and 4.22.

4.5 Almost and quasi- self-interlacing polynomials

In this section, we describe polynomials, which are dual (in the sense of Theorem 4.8) to the quasi-stable polynomials. Because of the mentioned duality we just give the definition of these polynomials.

Definition 4.23.

A polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ of degree $n$ is called quasi-self-interlacing of type I (of type II) with degeneracy index $m$ , $0\leqslant m\leqslant n$ , if the polynomial $p_{0}(-z^{2})-zp_{1}(-z^{2})$ (resp. the polynomial $p_{0}(-z^{2})+zp_{1}(-z^{2})$ ) is quasi-stable with degeneracy index $m$ .

In other words, the polynomial $p$ is quasi-self-interlacing if non-common zeroes f the polynomials $p(z)$ and $p(-z)$ are real, simple and interlacing.

Obviously, all results regarding quasi-stable polynomials can be easily reformulated for the quasi-stable polynomials. So we leave such reformulation to the reader.

In Section 5 we use the quasi-stable polynomials with degeneracy index $1$ , that is, the polynomials of the form $p(z)=zq(z)$ , where $q(z)$ is a self-interlacing polynomial. Such polynomials we call almost self-interlacing polynomials. Note that if $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ is almost self-interlacing, then the function $\Phi=\dfrac{p_{1}}{p_{0}}$ is an R-function with all poles positive except one, which is zero.

4.6 ”Strange” polynomials

This section is devoted to short description of some numeric results on the distribution of zeroes of some polynomials closely connected to the Hurwitz stable and self-interlacing polynomials. We call these polynomials ”strange” because of their curious zero location.

Let $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ be a Hurwitz stable polynomial of degree $n$ . Consider the polynomial $q(z)=p_{0}(-z^{2})+zp_{1}(z^{2})$ . It is evident that $q(0)=p(0)\neq 0$ . Calculations with Maple Software showed that the polynomial $q$ has exactly $\left[\dfrac{n+1}{2}\right]$ simple zeroes in the open right half-plane and exactly $\left[\dfrac{n}{2}\right]$ simple zeroes in the open left half-plane, but it has no zeroes on the imaginary axis. At least one zero of $q$ is nonreal.

Let us denote by $\mu_{1},\mu_{2},\ldots$ the distinct absolute values of the zeroes of the polynomial $q$ lying in the open left half-plane such that

0<\mu_{1}<\mu_{2}<\mu_{3}<\ldots

And denote by $\lambda_{1},\lambda_{2},\ldots$ the distinct absolute values of the zeroes of the polynomial $q$ lying in the open right half-plane such that

0<\lambda_{1}<\lambda_{2}<\lambda_{3}<\ldots

So numerical experiments showed that

0<\lambda_{1}<\mu_{1}<\lambda_{2}<\mu_{2}<\lambda_{3}<\ldots

The polynomial $p_{0}(z^{2})+zp_{1}(-z^{2})$ possesses similar properties according to calculations.

However, it is not known for sure if all the polynomials of the type $p_{0}(-z^{2})+zp_{1}(z^{2})$ , where the polynomial $p_{0}(z^{2})+zp_{1}(z^{2})$ is Hurwitz stable, have the zero location described above.

4.7 Matrices with self-interlacing spectrum

In this section, we consider some classes of real matrices with self-interlacing spectrum and develop a method of constructing such kind of matrices from a given totally positive matrix. But at first, we recall some definitions and statements from the book [10].

Definition 4.24 ([10]).

A square matrix $A=\|a_{ij}\|_{1}^{n}$ is called sign definite of class $n$ if for any $k\leqslant n$ , all the non-zero minors of order $k$ have the same sign $\varepsilon_{k}$ . The sequence $\{\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{n}\}$ is called the signature sequence of the matrix $A$ .

A sign definite matrix of class $n$ is called strictly sign definite of class $n$ if all its minors are different from zero.

Definition 4.25 ([10]).

A square sign definite matrix $A=\|a_{ij}\|_{1}^{n}$ of class $n$ is called the matrix of class $n^{+}$ if some its power is a strictly sign definite matrix of class $n$ .

Note that a sign definite (strictly sign definite) matrix of class $n$ with the signature sequence $\varepsilon_{1}=\varepsilon_{2}=\ldots=\varepsilon_{n}=1$ is totally nonnegative (strictly totally positive). Also a sign definite matrix of class $n^{+}$ with the signature sequence $\varepsilon_{1}=\varepsilon_{2}=\ldots=\varepsilon_{n}=1$ is an oscillating matrix (see [10]), that is, a totally nonnegative matrix whose certain power is strictly totally positive. It is clear from the Binet-Cauchy formula that the square of a sign definite matrix is totally nonnegative.

In [10] it was established the following lemma.

Lemma 4.26.

Let the matrix $A=\|a_{ij}\|_{1}^{n}$ be totally nonnegative. Then the matrices $B=\|a_{n-i+1,j}\|_{1}^{n}$ and $C=\|a_{i,n-j+1}\|_{1}^{n}$ are sign definite of class $n$ . Moreover, the signature sequence of the matrices $B$ and $C$ is as follows:

\varepsilon_{k}=(-1)^{\tfrac{k(k-1)}{2}},\quad k=1,2,\ldots,n.

(4.45)

Note that the matrices $B$ and $C$ can be represented as follows

B=JA,\qquad\text{and}\qquad C=AJ,

where

J=\begin{pmatrix}0&0&\dots&0&1\\ 0&0&\dots&1&0\\ \vdots&\vdots&\cdot&\vdots&\vdots\\ 0&1&\dots&0&0\\ 1&0&\dots&0&0\\ \end{pmatrix}.

(4.46)

It is easy to see that the matrix $J$ is sign definite of class $n$ (but not of class $n^{+}$ ) with the signature sequence of the form (4.45). So by the Binet-Cauchy formula [11] we obtain the following statement.

Theorem 4.27.

The matrix $A=\|a_{ij}\|_{1}^{n}$ is totally nonnegative if and only if the matrix $JA$ (or the matrix $AJ$ ) is sign definite of class $n$ with the signature sequence (4.45). The matrix $J$ is defined in (4.46).

Obviously, the converse statement is also true.

Theorem 4.28.

The matrix $A=\|a_{ij}\|_{1}^{n}$ is a sign definite matrix of class $n$ with the signature sequence (4.45) if and only if the matrix $JA$ (or the matrix $AJ$ ) is totally nonnegative.

In the sequel, we need the following two theorems established in the book [10].

Theorem 4.29.

Let the matrix $A=\|a_{ij}\|_{1}^{n}$ be a sign definite of class $n^{+}$ with the signature sequence $\varepsilon_{k}$ , $k=1,2,\ldots,n$ . Then all the eigenvalues $\lambda_{k}$ , $k=1,2,\ldots,n$ , of the matrix $A$ are nonzero real and simple, and if

|\lambda_{1}|>|\lambda_{2}|>\ldots>|\lambda_{n}|>0,

(4.47)

then

\emph{sign}\,\lambda_{k}=\dfrac{\varepsilon_{k}}{\varepsilon_{k-1}},\qquad k=1,2,\ldots,n,\,\varepsilon_{0}=1.

(4.48)

Theorem 4.30.

A totally nonnegative matrix $A=\|a_{ij}\|_{1}^{n}$ is oscillating if and only if $A$ is nonsingular and the following inequalities hold

a_{j,j+1}>0,\qquad\text{and}\qquad a_{j+1,j}>0,\qquad j=1,2,\ldots,n-1.

Let us introduce the following definition.

Definition 4.31.

A matrix $A$ is said to have the self-interlacing spectrum if its eigenvalues are real and simple and satisfy the following inequalities

\lambda_{1}>-\lambda_{2}>\lambda_{3}>\ldots>(-1)^{n-1}\lambda_{n}>0,

(4.49)

-\lambda_{1}>\lambda_{2}>-\lambda_{3}>\ldots>(-1)^{n}\lambda_{n}>0.

(4.50)

Now we are in a position to complement Lemma 4.26.

Theorem 4.32.

Let all the entries of a nonsingular matrix $A=\|a_{ij}\|_{1}^{n}$ be nonnegative¹³¹³13The matrices with nonnegative entries are usually called nonnegative matrices. and for each $i$ , $i=1,2,\ldots,n-1$ , there exist number $r_{1}$ and $r_{2}$ , $1\leqslant r_{1},r_{2}\leqslant n$ , such that

a_{n-i,r_{1}}\cdot a_{n+1-r_{1},i}>0,a_{n+1-i,r_{2}}\cdot a_{n+1-r_{2},i+1}>0

(4.51)

(\text{or}\qquad a_{i,n+1-r_{1}}\cdot a_{r_{1},n-i}>0,a_{i+1,n+1-r_{2}}\cdot a_{r_{2},n+1-i}>0).

(4.52)

The matrix $A$ is totally nonnegative if and only if the matrix $B=JA=\|b_{ij}\|_{1}^{n}$ (or, respectively, the matrix $C=AJ$ ) is sign definite of class $n^{+}$ with the signature sequence defined in (4.45). Moreover, the matrix $B$ (or, respectively, the matrix $C$ ) possesses a self-interlacing spectrum of the form (4.49).

Proof.

We prove the theorem in the case when the condition (4.51) holds. The case of the condition (4.52) can be established analogously.

Let $A$ be a nonsingular totally nonnegative matrix and the condition (4.51) holds. From Theorem 4.27 it follows that the matrix $B=JA$ is sign definite of class $n$ with the signature sequence (4.45). In order to the matrix be be sign definite of class $n^{+}$ it is necessary and sufficient that a certain power of this matrix $B$ be strictly sign definite of class $n$ . Since the entries of the matrix $J$ have the form

(J)_{ij}=\begin{cases}&1,\qquad i=n+1-j;\\ &0,\qquad i\neq n+1-j;\end{cases}

the entries of the matrix $B$ can be represented as follows:

b_{ij}=\sum\limits_{k=1}^{n}(J)_{ik}a_{kj}=a_{n+1-i,j}.

Consider the totally nonnegative matrix $B^{2}$ . Its entries have the form

(B^{2})_{ij}=\sum\limits_{k=1}^{n}b_{ik}b_{kj}=\sum\limits_{k=1}^{n}a_{n+1-i,k}a_{n+1-k,j}.

From these formulæ and from (4.51) it follows that all the entries of the matrix $B^{2}$ above and under the main diagonal are positive, that is, $(B^{2})_{i,i+1}>0$ $(B^{2})_{i+1,i}>0$ , $i=1,2,\ldots,n-1$ . By Theorem 4.30, $B^{2}$ is an oscillating matrix. According to the definition of oscillating matrices, a certain power of $B^{2}$ is strictly totally positive. Thus, we proved that a certain power of the matrix $B$ is strictly sign definite, so $B$ is a sign definite matrix of class $n^{+}$ with the signature sequence (4.45) according to Lemma 4.26. By Theorem 4.29, all eigenvalues of the matrix $B$ are nonzero real and simple. Moreover, if we enumerate the eigenvalues in order of decreasing absolute values as in (4.47), then from (4.48) and (4.45) we obtain that the spectrum of $B$ is of the form (4.49).

The converse assertion of the theorem follows from Theorem 4.27. ∎

Remark 4.33.

For $n=2l+1$ (for $n=2l$ ), the characteristic polynomial of the matrix $B$ described in Theorem 4.32 is self-interlacing of type I (of type II).

Remark 4.34.

If the matrix $-A$ is totally nonnegative and the conditions (4.51) (or the conditions (4.52)) hold, then the matrix $B=JA$ (or, respectively, $C=AJ$ ) has a spectrum of the form (4.50).

Remark 4.35.

One can obtain another types of totally nonnegative matrices which result matrices with self-interlacing spectra after multiplication by the matrix $J$ . To do this we need to change the conditions (4.51)–(4.52) by another ones such that, for instance, the matrix $B^{4}$ , or $B^{6}$ , (or $B^{8}$ etc.) becomes oscillating.

Theorem 4.32 implies the following corollary.

Corollary 4.36.

A nonsingular matrix $A$ with positive entries is totally nonnegative if and only if the matrix $B=JA$ (or the matrix $C=AJ$ ) is sign definite of class $n^{+}$ with the signature sequence (4.45). Moreover, the matrix $B$ has a self-interlacing spectrum of the form (4.49).

Consider a particular case of conditions (4.51). Suppose that all diagonal entries if the matrix $A$ be positive $a_{jj}>0$ , $j=1,2,\ldots,n$ . It is easy to see that in this case the conditions (4.51) hold if $a_{j,j+1}>0$ , $j=1,2,\ldots,n-1$ . If all remaining entries of the matrix $A$ are nonnegative, then by Theorem 4.32 the matrices $B=JA$ and $C=AJ$ have self-interlacing spectra if $A$ is totally nonnegative.

If all other entries of the matrix $A$ (that is, all entries except $a_{jj}$ and $a_{j,j+1}$ , which are positive) equal zero, then $A$ is a bidiagonal matrix with positive entries on and under the main diagonal. Clearly, $A$ is totally nonnegative. Then by Theorem 4.32 the matrix $B=JA$ has a self-interlacing spectrum. Thus we obtain the following statement established in [15]:

Theorem 4.37.

Any anti-bidiagonal $n\times n$ matrix with positive entries

B=\begin{pmatrix}0&0&0&\dots&0&0&b_{n}\\ 0&0&0&\dots&0&b_{n-2}&b_{n-1}\\ 0&0&0&\dots&b_{n-4}&b_{n-3}&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&c_{n-4}&\dots&0&0&0\\ 0&c_{n-2}&c_{n-3}&\dots&0&0&0\\ c_{n}&c_{n-1}&0&\dots&0&0&0\\ \end{pmatrix}

(4.53)

has a self-interlacing spectrum of the form (4.49). Here the entries $b_{j}>0$ , $j=2,3,\ldots,n$ , lie above the main diagonal, the entries $c_{j}>0$ , $j=2,3,\ldots,n$ , lie under the main diagonal and the only entry on the main diagonal is $a_{1}>0$ .

In [15] it was also proved that the spectrum of the matrix (4.53) coincides with the spectrum of the following tridiagonal matrix

K=\begin{pmatrix}a_{1}&b_{2}&0&\dots&0&0\\ c_{2}&0&b_{3}&\dots&0&0\\ 0&c_{3}&0&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&0&b_{n}\\ 0&0&0&\dots&c_{n}&0\\ \end{pmatrix}.

It is well-known [10] that the spectrum of this matrix does not depend on the entries $b_{j}$ and $c_{j}$ separately. It depends on products $b_{j}c_{j}$ , $j=2,3,\ldots,n$ . So in order to the matrices (4.53) and $K$ to have self-interlacing spectra, it is sufficient that the inequalities $a_{1}>0$ and $b_{j}c_{j}>0$ , $j=2,3,\ldots,n$ , hold.

Finally, consider a tridiagonal matrix

M_{J}=\begin{pmatrix}a_{1}&b_{1}&0&\dots&0&0\\ c_{1}&a_{2}&b_{2}&\dots&0&0\\ 0&c_{2}&a_{3}&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&a_{n-1}&b_{n-1}\\ 0&0&0&\dots&c_{n-1}&a_{n}\\ \end{pmatrix},

where $a_{k},b_{k},c_{k}\in\mathbb{R}$ and $c_{k}b_{k}\neq 0$ . In [10], there was proved the following fact.

Theorem 4.38.

The matrix $M_{J}$ is oscillating if and only if all the entries $b_{k}$ and $c_{k}$ are positive and all the leading principal minors of $M_{J}$ are also positive:

a_{1}>0,\;\begin{vmatrix}a_{1}&b_{1}\\ c_{1}&a_{2}\end{vmatrix}>0,\;\begin{vmatrix}a_{1}&b_{1}&0\\ c_{1}&a_{2}&b_{2}\\ 0&c_{2}&a_{3}\end{vmatrix}>0,\;\ldots\;\begin{vmatrix}a_{1}&b_{1}&0&\dots&0&0\\ c_{1}&a_{2}&b_{2}&\dots&0&0\\ 0&c_{2}&a_{3}&\dots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\dots&a_{n-1}&b_{n-1}\\ 0&0&0&\dots&c_{n-1}&a_{n}\\ \end{vmatrix}>0.

(4.54)

This theorem together with Theorem 4.32 implies the following statement.

Theorem 4.39.

The anti-tridiagonal matrix

A_{J}=\begin{pmatrix}0&0&0&\dots&0&b_{1}&a_{1}\\ 0&0&0&\dots&b_{2}&a_{2}&c_{1}\\ 0&0&0&\dots&a_{3}&c_{2}&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&b_{n-2}&a_{n-2}&\dots&0&0&0\\ b_{n-1}&a_{n-1}&c_{n-2}&\dots&0&0&0\\ a_{n}&c_{n-1}&0&\dots&0&0&0\\ \end{pmatrix},

where $a_{j},b_{j},c_{j}>0$ for $j=1,2,\ldots,n-1$ , is sign definite of class $n^{+}$ and has a self-interlacing spectrum (4.49) if and only if the following inequalities hold:

(-1)^{\frac{k(k-1)}{2}}A_{J}\begin{pmatrix}1&2&\dots&k\\ n+1-k&n+2-k&\dots&n\\ \end{pmatrix}>0,\quad k=1,2,\ldots,n.

(4.55)

Proof.

If the matrix $A_{J}$ is sign definite of class $n^{+}$ and has a self-interlacing spectrum of the form (4.49), then according to Theorem 4.29, the signs of its nonzero minors can be calculated by the formula (4.45). This implies the inequalities (4.55).

Conversely, let the inequalities (4.55) hold. Then we have that the inequalities (4.54) hold for the matrix $M_{J}=JA_{J}$ . By Theorem 4.38, the matrix $M_{J}$ is oscillating and, in particular, totally nonnegative. Now notice that $(M_{J})_{ii}>0$ , $i=1,\ldots,n$ , and $(M_{J})_{k,k+1}>0$ , $k=1,\ldots,k-1$ , so $M_{J}$ satisfies the condition (4.51) of Theorem 4.32. Therefore, the matrix $A_{J}$ is sign definite of class $n^{+}$ and has a self-interlacing spectrum of the form (4.49). ∎

5 Generalized Hurwitz polynomials

In this section, we describe the class of real polynomials whose associated function $\Phi$ is an R-function. In particular, this class includes all Hurwitz stable and self-interlacing polynomials as extremal cases.

5.1 General theory

Let us consider a polynomial

p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0.

(5.1)

Definition 5.1.

The polynomial $p$ is called generalized Hurwitz polynomial of type I of order $k$ , where $1\leqslant k\leqslant\left[\dfrac{n+1}{2}\right]$ , if it has exactly $k$ zeroes in the closed right half-plane, all of which are nonnegative and simple:

0\leqslant\mu_{1}<\mu_{2}<\cdots<\mu_{k},

(5.2)

Definition 5.2.

If $p(z)$ is a generalized Hurwitz polynomials of type I, then the polynomial $p(-z)$ is called generalized Hurwitz polynomial of type II.

It clear that all results obtained for the generalized Hurwitz polynomials of type I can be easily reformulated for the generalized Hurwitz polynomials of type I. Thus, in the rest of the section we consider only generalized Hurwitz polynomials of type I unless explicitly stipulated otherwise.

Since the generalized Hurwitz polynomials of order $k$ have exactly $k$ zeroes in the closed right half-plane, the generalized Hurwitz polynomials of order $0$ have no zeroes in the closed right half-plane, so they are Hurwitz stable.

Analogously, if $p$ is a generalized Hurwitz polynomial of order $k=\left[\dfrac{n+1}{2}\right]$ without a root at zero, then $p$ is self-interlacing of type I. In fact, let $0<\mu_{1}<\mu_{2}<\ldots<\mu_{k}$ be its positive zeroes. Then by definition, $p$ has an odd number of zeroes, counting multiplicities, (at least one) on each interval $(-\mu_{j+1},-\mu_{j})$ , $j=1,\ldots,k-1$ . Moreover, if $n=2l$ , then $p$ has at least one zero on the interval $(-\infty,-\mu_{k})$ . It is easy to see that $p$ can not have nonreal zeroes and has exactly one simple zero on each interval $(-\mu_{j+1},-\mu_{j})$ , $j=1,\ldots,k-1$ , and on the interval $(-\infty,-\mu_{k})$ for $n=2l$ . Besides, the zeroes of $p$ are distributed as in (4.1).

Thus, Hurwitz stable and self-interlacing polynomials are generalized Hurwitz polynomials of minimal and maximal orders, respectively. We note that generalized Hurwitz polynomials of maximal order with a root at zero are almost self-interlacing.

Now we are in a position to establish the main theorem of the theory of generalized Hurwitz polynomials. This theorem is a direct generalization of Theorems 3.3 and 4.3.

Theorem 5.3.

Let $p$ be a given real polynomial of degree $n\geqslant 1$ as in (5.1). The polynomial $p$ is generalized Hurwitz if and only if its associated function $\Phi$ defined in (2.7) is an R-function with exactly $l=\left[\dfrac{n}{2}\right]$ (for $a_{1}\neq 0$ ) or $l-1$ (for $a_{1}=0$ ) poles¹⁴¹⁴14Let us note that if $n=2l$ and $\Phi$ is an R-function, then $a_{1}\neq 0$ , so $\Phi$ has exactly $l$ poles.. Moreover,

•

if $n=2l$ or if $n=2l+1$ with

$a_{0}a_{1}>0,$ (5.3)

then the number of nonnegative poles of the function $\Phi$ equals order of $p$ ;
•

if $n=2l+1$ with

$a_{0}a_{1}\leqslant 0,$ (5.4)

then the number of nonnegative poles of the function $\Phi$ equals order of $p$ minus one.

Proof.

As we said above, for $k=0$ and $k=\left[\dfrac{n+1}{2}\right]$ this theorem is true.

Sufficiency. Let the function $\Phi$ be an R-function with exactly $r$ nonpositive poles, $0\leqslant r\leqslant\left[\dfrac{n}{2}\right]$ .

At first, assume that $a_{1}\neq 0$ and the function $\Phi$ has no pole at zero. Then $\Phi$ has exactly $l=\left[\dfrac{n}{2}\right]$ poles and can be represented in the following form by Theorem 1.18:

\Phi(u)=\alpha+\sum_{j=1}^{r}\frac{\alpha_{j}}{u-\omega_{j}}+\sum_{j=r+1}^{l}\frac{\alpha_{j}}{u+\omega_{j}},

(5.5)

where $\alpha\in\mathbb{R}$ ( $\alpha=0$ if and only if $n=2l$ ), $\alpha_{j}>0,\omega_{j}>0$ for $j=1,\ldots,l$ , all poles are distinct. Let the positive poles are enumerated as follows $0<\omega_{1}<\omega_{2}<\ldots<\omega_{r}$ .

As in the proofs of Theorems 3.3 and 4.3, we note that the set of zeroes of the polynomial $p$ coincides with the set of roots of the following equation

\Phi(z^{2})=-\dfrac{1}{z}.

(5.6)

So if we suppose that $p(\lambda)=0$ and $\operatorname{Im}\lambda\neq 0$ (that is, $\lambda$ is a nonreal zero of $p$ ), then from (5.6) we obtain

\displaystyle\frac{\operatorname{Im}\Phi(\lambda^{2})}{\operatorname{Im}(\lambda^{2})}=\frac{1}{|\lambda|^{2}}\frac{\operatorname{Im}\lambda}{\operatorname{Im}\lambda^{2}}=\frac{1}{2|\lambda|^{2}}\frac{1}{\operatorname{Re}\lambda}<0,

(5.7)

since $\Phi$ is an R-function by assumption. From (5.7) it follows that all nonreal zeroes of $p$ lie in the open left half-plane. Thus, all zeroes of $p$ are located in the open left half-plane or on the nonnegative real half-axis.

As in the proof of Theorem 4.3, we investigate the distribution of the real zeroes of the polynomial $p$ . By assumption, $|\Phi(0)|<\infty$ , so the polynomial $p$ does not vanish at zero. In fact, if $p(0)=0$ , then $p_{0}(0)=0$ , but it contradicts with $|\Phi(0)|<\infty$ . Thus, we suppose that $z\in\mathbb{R}\setminus\{0\}$ and change the variables as follows $z^{2}=u>0$ . Then for real nonzero $z$ , the equation (5.6) is equivalent to the following system

\begin{cases}\ \Phi(u)=-\dfrac{1}{\sqrt{u}},\\ \ \Phi(u)=\dfrac{1}{\sqrt{u}},\end{cases}\quad u>0.

(5.8)

We also note (as in the proof of Theorem 4.3) that all positive roots (if any) of the first equation (5.8) are squares of positive roots of the equation (5.6), and all positive roots (if any) of the second equation (5.8) are squares of negative roots of the equation (5.6). The equation (5.6) may have nonreal roots, but as we showed above all these roots are located in the open left half-plane.

Let $n=2l$ . Then $\lim\limits_{u\to\pm\infty}\Phi(u)=\alpha=0$ . The set of zeroes the function $F_{1}(u)=\Phi(u)+\dfrac{1}{\sqrt{u}}$ coincides with the set of roots of the first equation (5.8). In the same way as in the proof of Theorem 4.3, one can show that the function $F_{1}$ has exactly one simple zero, say $\mu_{i}^{2}$ ( $\mu_{i}>0$ ), on each interval $(\omega_{i-1},\omega_{i})$ , $i=2,\ldots,r$ and exactly one simple zero $\mu_{1}^{2}$ ( $\mu_{1}>0$ ) on the interval $(0,\omega_{1})$ . Moreover, denoting by $\xi_{i}$ a unique zero¹⁵¹⁵15Recall that $\Phi$ is an R-function by assumption. Consequently, it has exactly one simple zero on each interval $(\omega_{i},\omega_{i+1})$ . of the function $\Phi$ on the interval $(\omega_{i},\omega_{i+1})$ , $i=1,\ldots,r-1$ , we have

\mu_{i}^{2}<\omega_{i}<\xi_{i}<\mu_{i+1}^{2}<\omega_{i+1},\qquad i=1,\ldots,r-1,

(5.9)

since $F_{1}(\xi_{i})=\dfrac{1}{\sqrt{\xi_{i}}}>0$ , and $F_{1}$ is decreasing between its poles on the positive half-axis as a sum of decreasing functions.

Thus, the first equation (5.8) has exactly $r$ positive roots $\mu_{i}^{2}$ all of which are simple and satisfy the inequalities (5.9).

Now consider the function $F_{2}(u)=\Phi(u)-\dfrac{1}{\sqrt{u}}$ whose set of zeroes coincides with the set of roots of the second equation (5.8). In the same way as in the proof of Theorem 4.3, one can show that the function $F_{2}$ has an odd number of zeroes, counting multiplicities, on each interval $(\omega_{i},\mu^{2}_{i+1})$ , $i=1,\ldots,r-1$ , and on the interval $(\omega_{r},+\infty)$ . Moreover, $F_{2}(u)<0$ on the intervals $[\mu^{2}_{i+1},\omega_{i+1})$ , $i=1,\ldots,r-1$ . Let us now turn our attention to the interval $(0,\omega_{1})$ . Since $|\Phi(0)|<0$ by assumption and $-\dfrac{1}{\sqrt{u}}\to-\infty$ as $u\to+0$ , we have $F_{2}(u)\to-\infty$ as $u\to+0$ . Besides, $F_{2}(u)=F_{1}(u)-\dfrac{2}{\sqrt{u}}$ , so $F_{2}(\mu_{1}^{2})=-\dfrac{2}{\mu_{1}}<0$ . Therefore, $F_{2}$ has an even number of zeroes, counting multiplicities, on the interval $(0,\mu_{1}^{2})$ . However, since $F_{1}(u)$ is monotone decreasing on $(0,\omega_{1})$ as a sum of two monotone decreasing functions, we have $F_{1}(u)<0$ on the interval $(\mu_{1}^{2},\omega_{1})$ . Consequently, $F_{2}(u)$ is also negative for $u\in(\mu_{1}^{2},\omega_{1})$ , so its zeroes on the interval $(0,\omega_{1})$ (if any) lie, indeed, on the interval $(0,\mu_{1}^{2})$ .

Thus, we obtain that in the closed right half-plane, the polynomial $p$ has only $r$ zeroes, all of which are positive and simple:

0<\mu_{1}<\mu_{2}<\ldots<\mu_{r}.

Moreover, $p$ has an even number of zeroes, counting multiplicities, on the interval $(-\mu_{1},0)$ and an odd number of zeroes, counting multiplicities, on each interval $(-\mu_{i+1},-\mu_{i})$ , $i=1,\ldots,r-1$ . Also $p(-\mu_{i})\neq 0$ for $i=1,\ldots,r$ . As we showed above, all nonreal zeroes of $p$ are in the open left-half-plane, so $p$ is a generalized Hurwitz polynomial of order $r$ .

Let $n=2l+1$ , and the condition (5.3) holds. The only difference between this case and the case $n=2l$ is behaviour of the function $\Phi$ at infinity. Thus, as in the case $n=2l$ , $F_{1}$ has a unique simple zero, say $\mu_{i}^{2}$ ( $\mu_{i}>0$ ), $i=1,\ldots,r$ , on each interval $(0,\omega_{1})$ , $(\omega_{1},\omega_{2})$ , …, $(\omega_{r-1},\omega_{r})$ . These zeroes are distributed as in (5.9). Furthermore, we have $F_{1}(u)\to+\infty$ as $u\searrow\omega_{r}$ and $F_{1}(u)\to\alpha=\dfrac{a_{0}}{a_{1}}>0$ as $u\to+\infty$ , so $F_{1}(u)>0$ for $u\in(\omega_{r},+\infty)$ (recall that $\Phi$ is a monotone decreasing function on the interval $(\omega_{r},+\infty)$ and $\Phi(u)\to+\infty$ as $u\searrow\omega_{r}$ ). As in the case $n=2l$ , the function $F_{2}(u)$ has an even number of zeroes, counting multiplicities, on the interval $(0,\mu_{1}^{2})$ and an odd number of zeroes, counting multiplicities, on each interval $(\omega_{i},\mu^{2}_{i+1})$ for $i=1,\ldots,r-1$ . Moreover, $F_{2}$ has no zeroes on the intervals $(\mu_{1}^{2},\omega_{1})$ and $(\mu^{2}_{i+1},\omega_{i+1})$ , $i=1,\ldots,r-1$ . At the same time, we have $F_{2}(u)\to+\infty$ as $u\searrow\omega_{r}$ and $F_{2}(u)\to\alpha>0$ as $u\to+\infty$ . Since the function $F_{2}(u)$ is not monotone, it has an even number of zeroes, counting multiplicities, on the interval $(\omega_{r},+\infty)$ . Recall that these zeroes are squares of positive zeroes of the polynomial $p$ on the interval $(\omega_{r},+\infty)$ . Since the other zeroes of the functions $F_{1}$ and $F_{2}$ are distributed as in the case $n=2l$ , we obtain that $p$ is a generalized Hurwitz polynomial of order $r$ .

Let now $n=2l+1$ , and let $a_{0}a_{1}<0$ . This case also differs from the case $n=2l$ by behaviour of the function $\Phi$ at infinity: $\lim\limits_{u\to+\infty}\Phi(u)=\alpha=\dfrac{a_{0}}{a_{1}}<0$ . As in the case $n=2l$ , the function $F_{1}(u)=\Phi(u)+\dfrac{1}{\sqrt{u}}$ has a unique simple zero, say $\mu_{i}^{2}$ ( $\mu_{i}>0$ ), $i=1,\ldots,r$ , on each interval $(0,\omega_{1})$ , $(\omega_{1},\omega_{2})$ , …, $(\omega_{r-1},\omega_{r})$ . These zeroes are distributed as in (5.9). Furthermore, we have $F_{1}(u)\to+\infty$ as $u\searrow\omega_{r}$ and $F_{1}(u)\to\alpha<0$ as $u\to+\infty$ . Since $F_{1}(u)$ is decreasing on the interval $(\omega_{r},+\infty)$ as a sum of two decreasing functions on this interval, it has a unique simple zero, say $\mu_{r+1}^{2}$ ( $\mu_{r+1}>0$ ) on $(\omega_{r},+\infty)$ such that

\omega_{r}<\xi_{r}<\mu^{2}_{r+1},

where $\xi_{r}$ is a zero¹⁶¹⁶16Recall that this zero is unique, since $\Phi$ is a monotone function on $(\omega_{r},+\infty)$ . of $\Phi$ on the interval $(\omega_{r},+\infty)$ . As in the case $n=2l$ , the function $F_{2}(u)$ has an even number of zeroes, counting multiplicities, on the interval $(0,\mu_{1}^{2})$ and an odd number of zeroes, counting multiplicities, on each interval $(\omega_{i},\mu^{2}_{i+1})$ for $i=1,\ldots,r-1$ . Moreover, $F_{2}$ has no zeroes on the intervals $(\mu^{2}_{i},\omega_{i})$ , $i=1,\ldots,r$ . Now consider the interval $(\omega_{r},+\infty)$ . Since $F_{2}(u)\to+\infty$ as $u\searrow\omega_{r}$ and $F_{2}(\mu^{2}_{r+1})=-\dfrac{1}{\mu_{r+1}}<0$ , we obtain that $F_{2}$ has an odd number of zeroes, counting multiplicities, on the interval $(\omega_{r},\mu^{2}_{r+1})$ . Moreover, $F_{2}(u)=F_{1}(u)-\dfrac{2}{\sqrt{u}}<0$ for $(\mu^{2}_{r+1},+\infty)$ , because $F_{1}$ is negative on that interval. Since the positive roots of the first equation (5.8) are squares of the positive zeroes of the polynomial $p$ and the roots of the second equation (5.8) are squares of the negative zeroes of the polynomial $p$ , we obtain that that $p$ has only $r+1$ zeroes in the closed right half-plane, all of which are positive and simple:

0<\mu_{1}<\mu_{2}<\ldots<\mu_{r+1}.

Moreover, $p$ has an even number of zeroes, counting multiplicities, on the interval $(-\mu_{1},0)$ and an odd number of zeroes, counting multiplicities, on each interval $(-\mu_{i+1},-\mu_{i})$ , $i=1,\ldots,r$ . All nonreal zeroes of $p$ are in the open left half-plane, and $p(-\mu_{i})\neq 0$ for $i=1,\ldots,r+1$ , so $p$ is a generalized Hurwitz polynomial of order $r+1$ .

If $n=2l+1$ , and $a_{1}=0$ , then by Theorem 1.18, the function $\Phi$ has exactly $l-1$ poles and can be represented in the form

\Phi(u)=\beta u+\gamma+\sum_{j=1}^{r}\frac{\alpha_{j}}{u-\omega_{j}}+\sum_{j=r+1}^{l-1}\frac{\alpha_{j}}{u+\omega_{j}},

(5.10)

where $\beta<0$ , $\gamma\in\mathbb{R}$ , $\alpha_{j}>0,\omega_{j}>0$ for $j=1,\ldots,l$ , all poles are distinct. In the same way as in the case $n=2l+1$ with $a_{0}a_{1}<0$ , one can show that $p$ is a generalized Hurwitz polynomial of order $r+1$ , since $\Phi(u)$ is also negative for sufficiently large $u$ .

Suppose now that the function $\Phi$ has a pole at zero. Then $p_{0}(0)=0$ , so $p(0)=a_{n}=0$ . At the same time, $p_{1}(0)=a_{n-1}=p^{\prime}(0)\neq 0$ , since $\Phi$ is an R-function by assumption, so it has interlacing poles and zeroes (see Theorem 1.18). Thus, $p$ has a simple root at zero. Note that the set of zeroes of the polynomial $p$ , but the root at zero, coincides with the set of roots of the equation (5.6).

If $n=2l$ or $n=2l+1$ with (5.3), then as above, one can show that in the closed right half-plane $p$ has exactly $r$ zeroes, say $\mu_{i}$ , $i=1\ldots,r$ , all of which are nonnegative and simple:

0=\mu_{1}<\mu_{2}<\mu_{2}<\ldots<\mu_{r},

and $p(-\mu_{i})\neq 0$ for $i=2,\ldots,r$ . Also $p$ has an odd number of zeroes, counting multiplicities, on each interval $(-\mu_{i},-\mu_{i-1})$ and an odd (even) number of zeroes on the interval $(-\infty,-\mu_{r})$ if $n=2l$ ( $n=2l+1$ ). All nonreal zeroes of $p$ lie in the open left half-plane. Thus, $p$ is generalized Hurwitz of order $r$ .

If $n=2l+1$ and the condition (5.4) holds, then in the same way as above, one can show that $p$ is generalized Hurwitz of order $r+1$ .

Necessity. Let $p$ be a generalized Hurwitz polynomial of order $k$ . At first, suppose that $p(0)\neq 0$ .

Let $n=2l$ . Then by definition, $p(z)$ has only $k$ zeroes in the closed right half-plane, all of which are positive and simple: $0<\mu_{1}<\mu_{2}<\ldots<\mu_{k}$ , and $p(-\mu_{i})\neq 0$ . Moreover, $p$ has an odd number of zeroes, counting multiplicities, on each interval $(-\infty,-\mu_{k})$ , $(-\mu_{k},-\mu_{k-1})$ ,…, $(-\mu_{2},-\mu_{1})$ , and it has an even number of zeroes, counting multiplicities, on the interval $(-\mu_{1},0)$ . All nonreal zeroes of $p$ lie in the open left half-plane. Thus, we can represent the polynomial $p$ as a product $p(z)=h(z)g(z)$ , where $h$ is a self-interlacing polynomial¹⁷¹⁷17More exactly, $h$ is self-interlacing of type I., of degree $2k$ whose positive zeroes are $\mu_{i}$ , $i=1,\ldots,k$ , and $g$ is a Hurwitz stable polynomial of degree $2(l-k)$ , which has an even number of zeroes, counting multiplicities, on each interval $(-\infty,-\mu_{k})$ , $(-\mu_{k},-\mu_{k-1})$ ,…, $(-\mu_{2},-\mu_{1})$ , $(-\mu_{1},0)$ . Let

h(z)=h_{0}(z^{2})+zh_{1}(z^{2}),

g(z)=g_{0}(z^{2})+zg_{1}(z^{2}).

Then by Theorem 4.3, the function $H:=\dfrac{h_{1}}{h_{0}}$ is an R-function with exactly $k$ poles, all of which are positive. Analogously, by Theorem 3.3, the function $G:=\dfrac{g_{1}}{g_{0}}$ is an $R$ function with exactly $l-k$ poles, all of which are negative. If $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ , then

p_{1}(u)=h_{0}(u)g_{1}(u)+h_{1}(u)g_{0}(u),

(5.11)

p_{0}(w)=h_{0}(u)g_{0}(u)+uh_{1}(u)g_{1}(u).

(5.12)

Thus,

\displaystyle\Phi(u)=\frac{p_{1}(u)}{p_{0}(u)}=\frac{h_{0}(u)g_{1}(u)+h_{1}(u)g_{0}(u)}{h_{0}(u)g_{0}(u)+uh_{1}(u)g_{1}(u)}=\frac{H(u)+G(u)}{1+uH(u)G(u)}.

Note that the function

\displaystyle\frac{h_{1}(u)}{h_{0}(u)}+\frac{g_{1}(u)}{g_{0}(u)}=\frac{p_{1}(u)}{h_{0}(u)g_{0}(u)}

(5.13)

is an R-function as a sum of R-functions. Consequently, all zeroes of $p_{1}$ are real and simple and interlace zeroes of the polynomial $h_{0}g_{0}$ , which has $k$ positive simple zeroes and $l-k$ negative simple zeroes.

Furthermore, the function

\displaystyle F(u)=-\frac{h_{0}(u)}{h_{1}(u)}-\frac{ug_{1}(u)}{g_{0}(u)}=-\dfrac{1}{H(u)}-uG(u)=-\frac{p_{0}(u)}{h_{1}(u)g_{0}(u)}

(5.14)

is an R-function by Theorem 1.23, since the functions $H$ and $G$ are R-functions. Consequently, all zeroes of the polynomial $p_{0}$ are real and simple and interlace zeroes of the polynomial $h_{1}g_{0}$ (see Theorem 1.18). Thus, from (5.13)–(5.14) it follows that all poles and zeroes of the function $\Phi=\dfrac{p_{1}}{p_{0}}$ are real and simple. Therefore, $\Phi$ can be represented as follows

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=\sum\limits_{j=1}^{l}\dfrac{\alpha_{j}}{u-\omega_{j}},

(5.15)

where $\omega_{i}\in\mathbb{R}$ , $\omega_{i}\neq\omega_{j}$ for $i\neq j$ , and

\displaystyle\alpha_{j}=\frac{p_{1}(\omega_{j})}{p^{\prime}_{0}(\omega_{j})}\neq 0,\qquad j=1,2,\ldots,l.

According to Theorem 1.18, now it is sufficient to establish positivity of the numbers $\alpha_{j}$ to prove that $\Phi$ is an R-function.

Let $\omega_{j}$ be a zero of the polynomial $p_{0}$ . Since $p(0)\neq 0$ by assumption, we have $p_{0}(0)\neq 0$ , so $\omega_{j}\neq 0$ , $j=1,2,\ldots,l$ . It is clear that $h_{1}(\omega_{j})g_{0}(\omega_{j})\neq 0$ , because the zeroes of the polynomial $p_{0}$ interlace the zeroes of $h_{1}g_{0}$ . From (5.12) it follows that

\displaystyle h_{0}(\omega_{j})=-\frac{\omega_{j}h_{1}(\omega_{j})g_{1}(\omega_{j})}{g_{0}(\omega_{j})},

\begin{array}[]{l}\displaystyle p_{1}(\omega_{j})=h_{1}(\omega_{j})g_{0}(\omega_{j})-\frac{\omega_{j}h_{1}(\omega_{j})g_{1}^{2}(\omega_{j})}{g_{0}(\omega_{j})}=\\ =\dfrac{h_{1}(\omega_{j})}{g_{0}(\omega_{j})}\left[g_{0}^{2}(\omega_{j})-\omega_{j}g_{1}^{2}(\omega_{j})\right]=\dfrac{h_{1}(\omega_{j})}{g_{0}(\omega_{j})}g(\sqrt{\omega_{j}})g(-\sqrt{\omega_{j}}).\end{array}

According to (5.14), we have $p_{0}(u)=-h_{1}(u)g_{0}(u)F(u)$ , and $F(\omega_{j})=0$ , therefore

p_{0}^{\prime}(\omega_{j})=-h_{1}(\omega_{j})g_{0}(\omega_{j})F^{\prime}(\omega_{j}).

So we obtain

\alpha_{j}=\dfrac{p_{1}(\omega_{j})}{p^{\prime}_{0}(\omega_{j})}=\dfrac{h_{1}(\omega_{j})}{g_{0}(\omega_{j})}\cdot\dfrac{g(\sqrt{\omega_{j}})g(-\sqrt{\omega_{j}})}{-h_{1}(\omega_{j})g_{0}(\omega_{j})F^{\prime}(\omega_{j})}=-\dfrac{g(\sqrt{\omega_{j}})g(-\sqrt{\omega_{j}})}{g_{0}^{2}(\omega_{j})F^{\prime}(\omega_{j})}.

Since $F$ is an R-function, we have $F^{\prime}(u)<0$ , so $g_{0}^{2}(\omega_{j})F^{\prime}(\omega_{j})<0$ . Thus, we obtain that

\textrm{sign}(\alpha_{j})=\textrm{sign}\left(g(\sqrt{\omega_{j}})g(-\sqrt{\omega_{j}})\right).

If $\omega_{j}<0$ , then $g(\sqrt{\omega_{j}})g(-\sqrt{\omega_{j}})=g_{0}^{2}(\omega_{j})-\omega_{j}g_{1}^{2}(\omega_{j})>0$ , since $g_{0}$ and $g_{1}$ are real polynomials. So $\alpha_{j}>0$ in this case.

Let now $\omega_{j}>0$ . The polynomial $g$ is Hurwitz stable with positive leading coefficient by construction, so $g(z)>0$ for all $z\geqslant 0$ . Therefore, $\textrm{sign}(\alpha_{j})=\textrm{sign}\left(g(-\sqrt{\omega_{j}})\right)$ . Since $\deg p_{0}=l$ , $\deg g_{0}=l-k$ , $\deg h_{1}=k-1$ , the function $F$ defined in (5.14) has exactly $l$ zeroes and $l-1$ poles. Consequently, the maximal positive zero of the polynomial $p_{0}$ is greater than the maximal positive zero of the polynomial $h_{1}$ (recall that all zeroes of $h_{1}$ are positive). Besides, $F$ is decreasing between its poles, so by (5.12), we obtain $F(0)=-\dfrac{p_{0}(0)}{h_{1}(0)g_{0}(0)}=-\dfrac{h_{0}(0)}{h_{1}(0)}>0$ , since $h$ is self-interlacing by construction, and (4.20) holds for it. This means that the function $F$ (and the polynomial $p_{0}$ ) has a positive zero, which is less than the minimal positive zero of $h_{1}$ . Thus, the polynomial $p_{0}$ has exactly $k$ positive zeroes and, respectively, exactly $l-k$ negative zeroes.

Let us enumerate the positive zeroes of $p_{0}$ as follows

0<\omega_{1}<\omega_{2}<\ldots<\omega_{k}.

Then denoting the zeroes of the polynomial $h_{1}$ by $\gamma_{j}$ $(j=1,2,\ldots,k-1)$ , we obtain

0<\omega_{1}<\gamma_{1}<\omega_{2}<\gamma_{2}<\ldots<\omega_{k}<\gamma_{k-1}<\omega_{k}.

Let us now show that $\omega_{1}>\mu_{1}^{2}$ . Indeed, since $\mu_{1}$ is the minimal positive zero of $h$ and $h(0)\neq 0$ , the polynomial $h$ does not change its sign on the interval $[0,\mu_{1})$ . Moreover, both polynomials $h_{0}$ and $h_{1}$ have no zeroes on the interval $[0,\mu_{1}^{2}]$ (see the proof of Theorem 4.3). Consequently, we get $\textrm{sign}(h(0))=\textrm{sign}\left(h_{0}(0)\right)=-\textrm{sign}\left(h_{1}(0)\right)=\textrm{sign}\left(h_{0}(\mu_{1}^{2})\right)=-\textrm{sign}\left(h_{1}(\mu_{1}^{2})\right)$ that follows from the self-interlacing of $h$ . Note that $\textrm{sign}\left(p_{0}(0)\right)=\textrm{sign}\left(h_{0}(0)g_{0}(0)\right)=\textrm{sign}\left(h_{0}(0)\right)$ , since $g$ is Hurwitz stable with the positive leading coefficient, so $g_{0}(0)>0$ . The number $\mu_{1}$ is a zero of the polynomial $h$ , therefore $h_{0}(\mu_{1}^{2})=-\mu_{1}h_{1}(\mu_{1}^{2})$ , so from (5.12) we have

p_{0}(\mu_{1}^{2})=h_{0}(\mu_{1}^{2})g_{0}(\mu_{1}^{2})-\mu_{1}h_{0}(\mu_{1}^{2})g_{1}(\mu_{1}^{2})=h_{0}(\mu_{1}^{2})g(-\mu_{1}).

By construction, the polynomial $g$ has an even number of zeroes, counting multiplicities, on each interval $(-\infty,-\mu_{k}),(-\mu_{k},-\mu_{k-1}),\ldots,(-\mu_{2},-\mu_{1}),(-\mu_{1},0)$ , therefore $\textrm{sign}\left(g(0)\right)=\textrm{sign}\left(g(-\mu_{j})\right)$ , $j=1,2,\ldots,k$ , that is, $g(-\mu_{j})>0$ . This implies that $\textrm{sign}\left(p_{0}(\mu_{1}^{2})\right)=\textrm{sign}\left(h_{0}(\mu_{1}^{2})\right)=\textrm{sign}\left(h_{0}(0)\right)=\textrm{sign}\left(p_{0}(0)\right)$ . Consequently, the polynomial $p_{0}$ has an even number of zeroes, counting multiplicities, on the interval $(0,\mu_{1}^{2})$ . But the positive zeroes of the polynomials $p_{0}$ and $h_{1}$ interlace, and $h_{1}$ has no zeroes in the interval $(0,\mu_{1}^{2})$ , therefore $p_{0}$ has also no zeroes on that interval, that is, $\omega_{1}>\mu_{1}^{2}$ .

Note that (see the proof of Theorem 4.3)

0<\mu_{1}^{2}<\gamma_{1}<\mu_{2}^{2}<\gamma_{2}<\ldots<\mu_{k-1}^{2}<\gamma_{k-1}<\mu_{k}^{2}.

Thus, we already established that $0<\mu_{1}^{2}<\omega_{1}<\gamma_{1}<\mu_{2}^{2}$ . Now we prove that the polynomial $p_{0}$ has no zeroes on each interval $(\gamma_{j},\mu_{j+1}^{2})$ , $j=1,2,\ldots,k-1$ . In fact, since $\textrm{sign}\left(p_{0}(0)\right)=\textrm{sign}\left(h_{0}(0)\right)$ and both polynomials $p_{0}$ and $h_{0}$ change their signs on each interval $(\gamma_{j},\gamma_{j+1})$ , $j=1,2,\ldots,k-1$ , we obtain $\textrm{sign}\left(p_{0}(\gamma_{j})\right)=\textrm{sign}\left(h_{0}(\gamma_{j})\right)$ for all $j=1,2,\ldots,k-1$ . By Theorem 4.3, the following inequalities hold

0<\mu_{1}^{2}<\beta_{1}<\gamma_{1}<\mu_{2}^{2}<\beta_{2}<\gamma_{2}<\ldots<\mu_{k-1}^{2}<\beta_{k-1}<\gamma_{k-1}<\mu_{k}^{2}<\beta_{k},

(5.16)

where $\beta_{j}$ are the zeroes of the polynomial $h_{0}$ . From (5.16) it follows that $h_{0}$ does not change its sign on each interval $(\gamma_{j},\mu_{j+1}^{2})$ , $j=1,2,\ldots,k-1$ , that is, $\textrm{sign}\left(h_{0}(\gamma_{j})\right)=\textrm{sign}\left(h_{0}(\mu_{j+1}^{2})\right)$ for all $j=1,2,\ldots,k-1$ , in particular. Since $h(\mu_{j+1})=0$ and therefore $h_{0}(\mu_{j+1}^{2})=-\mu_{j}h_{1}(\mu_{j+1}^{2})$ , we have

p_{0}(\mu_{j+1}^{2})=h_{0}(\mu_{j+1}^{2})g_{0}(\mu_{j+1}^{2})-\mu_{j+1}h_{0}(\mu_{j+1}^{2})g_{1}(\mu_{j+1}^{2})=h_{0}(\mu_{j+1}^{2})g(-\mu_{j+1}).

The inequalities $g(-\mu_{j+1})>0$ , $j=1,\ldots,k-1$ , proved above imply $\textrm{sign}\left(p_{0}(\mu_{j+1}^{2})\right)=\textrm{sign}\left(h_{0}(\mu_{j+1}^{2})\right)=\textrm{sign}\left(h_{0}(\gamma_{j})\right)=\textrm{sign}\left(p_{0}(\gamma_{j})\right)$ . Consequently, the polynomial $p_{0}$ has an even number of zeroes, counting multiplicities, on each interval $(\gamma_{j},\mu_{j+1}^{2})$ , $j=1,\ldots,k-1$ . But the zeroes of the polynomials $p_{0}$ and $h_{1}$ interlace, and $h_{1}$ has no zeroes on the intervals $(\gamma_{j},\mu_{j+1}^{2})$ , $j=1,\ldots,k-1$ , therefore $p_{0}$ also has no zeroes on those intervals. So we proved the following inequalities

\mu_{j}^{2}<\omega_{j}<\gamma_{j}<\mu_{j+1}^{2}\qquad\text{for}\qquad j=1,\ldots,k-1.

(5.17)

Let us consider the intervals $(\mu_{j}^{2},\omega_{j})$ , $j=1,\ldots,k$ and prove that the polynomial $g(-\sqrt{u})$ has no zeroes on those interval. Suppose that it is not true. Then there exists a number $\beta\in(\mu_{j}^{2},\omega_{j})$ such that $g(-\sqrt{\beta})=0$ . Then $g_{0}(\beta)=\sqrt{\beta}g_{1}(\beta)>0$ , and from the equality

p_{0}(\beta)=h_{0}(\beta)g_{0}(\beta)+\beta h_{1}(\beta)g_{1}(\beta)=g_{0}(\beta)h(\sqrt{\beta}),

we obtain that $\textrm{sign}\left(p_{0}(\beta)\right)=\textrm{sign}\left(h(\sqrt{\beta})\right)$ . It contradicts the equality $\textrm{sign}\left(p_{0}(\beta)\right)=-\textrm{sign}\left(h(\sqrt{\beta})\right)$ , which follows from (5.17) and from the equalities $\textrm{sign}\left(p_{0}(0)\right)=\textrm{sign}\left(h_{0}(0)\right)=\textrm{sign}\left(h(0)\right)$ established above.

Thus, we obtain that the polynomial $g$ has real zeroes (en even number of zeroes, counting multiplicities) only on the intervals $(-\infty,\sqrt{\omega_{k}})$ , $(-\mu_{k},-\sqrt{\omega_{k-1}})$ , …, $(-\mu_{2},-\sqrt{\omega_{1}}),(-\mu_{1},0)$ . Consequently, the following equalities hold $\textrm{sign}\left(g(0)\right)=\textrm{sign}\left(g(-\mu_{j})\right)=\textrm{sign}\left(g(-\sqrt{\omega_{j}})\right)=\textrm{sign}(\alpha_{j})>0$ for all $j=1,2,\ldots,k$ . So we showed that if $n=2l$ and $p(0)\neq 0$ , then the function $\Phi(u)=\dfrac{p_{1}}{p_{0}}$ can be represented as in (5.15) with positive $\alpha_{j}$ and with exactly $k$ positive poles, as required.

Let now $n=2l+1$ and $p(0)\neq 0$ . Then by definition, the polynomial $p$ has only $k$ zeroes in the closed right half-plane, all of which are positive: $0<\mu_{1}<\mu_{2}<\ldots<\mu_{k}$ . Moreover, $p$ has an odd number of zeroes of $p$ , counting multiplicities, on each interval $(-\mu_{k},-\mu_{k-1})$ , …, $(-\mu_{3},-\mu_{2})$ , $(-\mu_{2},-\mu_{1})$ , and an even number of zeroes, counting multiplicities, on the intervals $(-\infty,-\mu_{k})$ and $(-\mu_{1},0)$ . All nonreal zeroes of $p$ are located in the open left half-plane.

As in the case $n=2l$ , we represent the polynomial $p$ as a product $p(z)=g(z)h(z)$ , where $h$ is self-interlacing polynomial (of type I) of degree $2k-1$ whose positive zeroes are $\mu_{1},\mu_{2},\ldots,\mu_{k}$ , and $g$ is Hurwitz stable polynomial of degree $2(l+1-k)$ , which has an even number of zeroes, counting multiplicities, on each interval $(-\infty,-\mu_{k}),(-\mu_{k},-\mu_{k-1}),\ldots$ , $(-\mu_{2},-\mu_{1}),(-\mu_{1},0)$ . In the same way as above one can prove that the function $\Phi=\dfrac{p_{1}}{p_{0}}$ is an R-function. The only difference between this case and the case of $n=2l$ is the number of positive poles of the function $\Phi$ . In fact, let us consider the function $F$ defined in (5.14). In this case, we have $\deg p_{0}=l$ if $a_{1}\neq 0$ (or $\deg p_{0}=l-1$ if $a_{1}=0$ ), $\deg h_{1}=k-1$ , $\deg g_{0}=l+1-k$ , therefore $\deg(h_{1}g_{0})=l$ . Thus, the number of poles of the function $F$ is equal to (or greater than) the number of its zeroes.

Let

g(z)=b_{0}z^{2(l-k)+2}+c_{1}z^{2(l-k)+1}+\ldots,

h(z)=c_{0}z^{2k-1}+c_{1}z^{2k-2}+\ldots

Then the leading coefficient of the polynomial $h_{1}$ equals $c_{0}$ , and the leading coefficient of the polynomial $g_{0}$ equals $b_{0}$ according to (2.4)–(2.5). Consequently, the leading coefficient of the polynomial $h_{1}g_{0}$ is equal to $c_{0}b_{0}=a_{0}$ . Moreover, the leading coefficient of the polynomial $p_{0}$ is equal to $a_{1}$ or¹⁸¹⁸18If $a_{1}=0$ . $a_{3}$ by (2.5), so we have $\displaystyle\lim_{u\to\infty}F(u)=-\frac{a_{1}}{a_{0}}$ or $-\dfrac{a_{3}}{a_{0}}$ . The function $F$ is an R-function (that can be proved in the same way as above), so it is decreasing between its poles, and $F(0)=-\dfrac{h_{0}(0)}{h_{1}(0)}>0$ (see (4.20)). Therefore, the minimal positive zero of the polynomial $p_{0}$ is less than the minimal positive zero of the polynomial $h_{1}$ .

If the condition (5.3) holds, then $\displaystyle\lim_{u\to\infty}F(u)=-\dfrac{a_{1}}{a_{0}}<0$ , so the maximal positive zero of $p_{0}$ (the maximal positive zero of the function $F$ ) is greater than the maximal positive zero of the polynomial $h_{1}$ . This means that the polynomial $p_{0}$ has exactly $\deg h_{1}+1=k$ positive zeroes, so the function $\Phi$ has exactly $k$ positive poles.

If the condition (5.4) holds, then either $F(u)\to+0$ as $u\to+\infty$ if $a_{1}=0$ or $F(u)\to-\dfrac{a_{1}}{a_{0}}>0$ as $u\to+\infty$ if $a_{1}\neq 0$ , so we have $\displaystyle\lim_{u\to+\infty}F(u)>0$ . This implies that the maximal positive zero of $h_{1}$ is greater than the maximal positive zero of $p_{0}$ , so the polynomial $p_{0}$ has exactly $k-1=\deg h_{1}$ positive zeroes. Consequently, $\Phi$ has exactly $k-1$ positive poles. Thus, for $n=2l+1$ and $p(0)\neq 0$ , we obtain that the function $\Phi=\dfrac{p_{1}}{p_{0}}$ can be represented as in (5.15) with positive $\alpha_{j}$ and with exactly $k$ (if (5.3) holds) or $k-1$ (if (5.4) holds) positive poles, as required.

Let $n=2l$ and $p(0)=0$ . Then by definition, the polynomial $p$ has exactly $k$ zeroes in the closed right half-plane: $0=\mu_{1}<\mu_{2}<\ldots<\mu_{k}$ . Moreover, $p$ has an odd number of zeroes on each interval $(-\infty,-\mu_{k})$ , $(-\mu_{k},-\mu_{k-1})$ , …, $(-\mu_{3},-\mu_{2})$ , $(-\mu_{2},0)$ , counting multiplicities. All nonreal zeroes of $p$ are located in the open left half-plane.

As above, we represent the polynomial $p$ as a product: $p(z)=g(z)h(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ , where $h(z)=h_{0}(z^{2})+zh_{1}(z^{2})$ is an almost self-interlacing polynomial¹⁹¹⁹19See Section 4.5. (of type I) of degree $2k$ whose nonnegative zeroes are $\mu_{1},\mu_{2},\ldots,\mu_{k}$ , and $g(z)=g_{0}(z^{2})+zg_{1}(z^{2})$ is a Hurwitz stable polynomial of degree $2(l-k)$ , which has an even number of zeroes, counting multiplicities, on each interval $(-\infty,-\mu_{k})$ , $(-\mu_{k},-\mu_{k-1})$ , …, $(-\mu_{3},-\mu_{2})$ , $(-\mu_{2},0)$ .

In the same way as above, one can establish that $\Phi$ is an R-function. However, $p_{0}$ has a root at zero in this case. The polynomial $h_{0}$ also has a root at zero, and all its other roots are positive. Since $\deg h=2k$ , we have $\deg h_{0}=\deg h_{1}+1=k$ . Besides, the zeroes of $h_{0}$ interlace the zeroes of $h_{1}$ , so all zeroes of the polynomial $h_{1}$ are positive. Consequently, the R-function $F$ defined in (5.14) has $k-1$ positive poles and $l-k$ negative poles. It is easy to see that $\deg(g_{0}h_{1})=l-1$ and $\deg p_{0}=l$ , so the maximal positive zero of the polynomial $p_{0}$ (and of the function $F$ ) is greater than the maximal positive zero of the polynomial $h_{1}$ . Also the minimal positive of $p_{0}$ is also greater than the minimal positive zero of $h_{1}$ , since $F$ is decreasing between its poles and $F(0)=0$ . Thus, we obtain that $p_{0}$ has exactly $k-1$ positive zeroes, since the zeroes of $p_{0}$ interlace the zeroes of $g_{0}h_{1}$ . Consequently, the function $\Phi$ has $l-k$ negative poles, $k-1$ positive poles and a pole at zero.

If $n=2l+1$ and the polynomial $p(z)=p_{0}(z^{2})+zp_{1}(z^{2})$ has a root at zero, then in the same as above, one can prove that $\Phi$ is an R-function with either exactly $k-1$ positive poles (if the condition (5.3) holds), or exactly $k-2$ positive poles (if the condition (5.4) holds), since we have $F(0)=0$ for the R-function $F$ defined in (5.14). ∎

Theorem 5.3 allow us to use properties of R-functions to obtain additional criteria for polynomials from the class of generalized the Hurwitz polynomials. In particular, we can generalize Hurwitz and Liénard and Chipart criteria.

Let us consider the Hankel matrix $S(\Phi)=\|s_{i+j}\|^{\infty}_{i,j=0}$ constructed with the coefficients of the series (2.13) or (2.18). From Theorems 5.3, 1.18, and 1.24 we obtain the following criteria for generalized Hurwitz polynomials.

Theorem 5.4.

The polynomial $p$ defined in (5.1), with $a_{1}\neq 0$ if $n=2l+1$ , is generalized Hurwitz if and only if

D_{j}(\Phi)>0,\qquad j=1,\ldots,l.

(5.18)

Its order $k$ equals:

1)

if $n=2l$ , then

$k=l-\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi)),$ (5.19)
2)

if $n=2l+1$ , then

$k=l-\operatorname{{\rm V}}(-s_{-1},1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))+1,$ (5.20)

where $r=l$ for $\widehat{D}_{l}(\Phi)\neq 0$ , and $r=l-1$ for²⁰²⁰20By Corollary 1.5, if $\widehat{D}_{l}(\Phi)=0$ , then $\widehat{D}_{l-1}(\Phi)\neq 0$ . $\widehat{D}_{l}(\Phi)=0$ . Here $s_{-1}=\dfrac{a_{0}}{a_{1}}$ .

The number of sign changes in the sequence $1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi)$ must be calculated according to the Frobenius rule provided by Theorem 1.6.

Proof.

According to Theorem 5.3, the polynomial $p$ , with $a_{1}\neq 0$ if $n=2l+1$ , is generalized Hurwitz if and only if its associated function $\Phi$ is an R-function with exactly $l=\left[\dfrac{n}{2}\right]$ poles. But by Theorem 1.18, this is equivalent to the inequalities (5.18).

If $n=2l$ , or if $n=2l+1$ with (5.3), then by Theorem 5.3 the number of nonnegative poles of the function $\Phi$ is equal to order of the polynomial $p$ . From Theorem 1.24 we obtain (5.19), which is equivalent to (5.20) if $n=2l+1$ and (5.3) holds.

If $n=2l+1$ and $a_{0}a_{1}<0$ , then by Theorem 5.3, the number of nonnegative poles of $\Phi$ is equal to order of $p$ minus one. Thus, by Theorem 1.24 we have

k-1=l-\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi)),

that is equivalent to (5.20) in this case. ∎

In the same way as above one can establish the corresponding theorem for polynomials of odd degree with $a_{1}=0$ .

Theorem 5.5.

Let the polynomial $p$ of degree $n=2l+1$ be defined in (5.1) and let $a_{1}=0$ . The polynomial $p$ is generalized Hurwitz if and only if $a_{0}a_{3}<0$ and

D_{j}(\Phi)>0,\qquad j=1,\ldots,l-1.

(5.21)

Its order $k$ equals:

k=l-\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi)),

(5.22)

where $r=l-1$ for $\widehat{D}_{l-1}(\Phi)\neq 0$ , and $r=l-2$ for²¹²¹21By Corollary 1.5, if $\widehat{D}_{l-1}(\Phi)=0$ , then $\widehat{D}_{l-2}(\Phi)\neq 0$ . $\widehat{D}_{l-1}(\Phi)=0$ .

The number of sign changes in the sequence $1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi)$ must be calculated according to the Frobenius rule provided by Theorem 1.6.

Proof.

In fact, let the polynomial $p$ is generalized Hurwitz of order $k$ . Then by Theorem 5.3, the function $\Phi=\dfrac{p_{1}}{p_{0}}$ is an R-function. In particular, this means that $|\deg p_{0}-\deg p_{1}|\leqslant 1$ (see e.g. [7, 16]), therefore, $\deg p_{1}=l$ and $\deg p_{0}=l-1$ , since $\deg p=2l+1$ and $a_{1}=0$ by assumption. So by Theorems 1.18 and 5.3 the function $\Phi$ has the form

\Phi(u)=\dfrac{a_{0}}{a_{3}}\,u+\dfrac{a_{2}a_{3}-a_{0}a_{5}}{a_{3}^{2}}+\sum\limits_{j=1}^{k-1}\dfrac{\alpha_{j}}{u-\omega_{j}}+\sum\limits_{j=k}^{l-1}\dfrac{\alpha_{j}}{u+\nu_{j}},

(5.23)

where $\dfrac{a_{0}}{a_{3}}<0$ , $\alpha_{j}>0$ for $j=1,\ldots,l-1$ , $\omega_{j}\geqslant 0$ , $j=1\ldots,k-1$ , and $\nu_{j}>0$ for $j=k,\ldots,l-1$ . The inequalities (5.21) and the formula (5.22) now follow from Theorems 1.18 and 1.24.

Conversely, from the conditions (5.21)–(5.22), $a_{1}=0$ and from the inequality $a_{0}a_{3}<0$ it follows that the function $\Phi$ has the form (5.23) by Theorems 1.18 and 1.24. Now Theorem 5.3 implies that $p$ is a generalized Hurwitz polynomial of order $k$ . ∎

From Theorems 5.4 and 5.5 and from the formulæ (2.14)–(2.15) and (2.21) we obtain the following theorem, which is an analogue of Hurwitz stability criterion for generalized Hurwitz polynomials.

Theorem 5.6 (Generalized Hurwitz theorem).

The polynomial $p$ given in (5.1) is generalized Hurwitz if and only if

\Delta_{n-1}(p)>0,\ \Delta_{n-3}(p)>0,\ \Delta_{n-5}(p)>0,\ldots

(5.24)

The order $k$ of the polynomial $p$ equals

k=\operatorname{{\rm V}}(\Delta_{n}(p),\Delta_{n-2}(p),\ldots,1)\qquad\text{if}\quad p(0)\neq 0,

(5.25)

k=\operatorname{{\rm V}}(\Delta_{n-2}(p),\Delta_{n-4}(p),\ldots,1)+1\qquad\text{if}\quad p(0)=0.

(5.26)

Here the number of sign changes in (5.25) and (5.26) must be calculated according to the Frobenius rule provided by Theorem 1.6.

Proof.

Let the real polynomial $p$ be of degree $n$ such that the coefficient $a_{1}$ is nonzero for $n=2l+1$ . By Theorem 5.4, the polynomial $p$ is generalized Hurwitz if and only if the inequalities (5.18) hold. These inequalities are equivalent to the inequalities (5.24) according to the formulæ (2.14)–(2.15).

If the polynomial $p$ is of odd degree and $a_{1}=0$ , then by Theorem 5.5, the polynomial $p$ is generalized Hurwitz if and only if the inequalities (5.21) and $-a_{0}a_{3}>0$ hold. Those inequalities are equivalent to the inequalities (5.24) according to the formulæ (2.21)–(2.22).

We now prove that order of the polynomial can be calculated by the formulæ (5.25)–(5.26).

Let $n=2l$ , then from (5.19) and (2.14) we obtain

\begin{array}[]{c}k=l-\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))=l-\operatorname{{\rm V}}(1,-\Delta_{2}(p),\Delta_{4}(p),\ldots,(-1)^{r}\Delta_{2r}(p))=\\ \\ =\operatorname{{\rm V}}(1,\Delta_{2}(p),\Delta_{4}(p),\ldots,\Delta_{2r}(p))+l-r.\end{array}

This is exactly (5.25) for $r=l$ and exactly (5.26) for $r=l-1$ .

Let now $n=2l+1$ and $a_{0}a_{1}>0$ . The formulæ (5.20) and (2.14) imply

\begin{array}[]{c}k=l-\operatorname{{\rm V}}(-s_{-1},1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))+1=\\ \\ =l+1-\operatorname{{\rm V}}(-s_{-1},1)+\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))=\\ \\ =l-\operatorname{{\rm V}}(1,-a_{1}^{-3}\Delta_{3}(p),a_{1}^{-5}\Delta_{5}(p),\ldots,(-1)^{r}a_{1}^{-2r-1}\Delta_{2r+1}(p))=\\ \\ =\operatorname{{\rm V}}(1,\Delta_{1}(p),\Delta_{3}(p),\Delta_{5}(p),\ldots,\Delta_{2r+1}(p))+l-r,\end{array}

(5.27)

since $a_{1}=\Delta_{1}(p)>0$ . One can see that (5.27) implies (5.25) for $r=l$ and (5.26) for $r=l-1$ .

Let $n=2l+1$ and $a_{0}a_{1}<0$ . The formulæ (5.20) and (2.14) imply

\begin{array}[]{c}k=l-\operatorname{{\rm V}}(-s_{-1},1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))+1=\\ \\ =l-\operatorname{{\rm V}}(-s_{-1},1)+\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))+1=\\ \\ =1+l-\operatorname{{\rm V}}(1,-a_{1}^{-3}\Delta_{3}(p),a_{1}^{-5}\Delta_{5}(p),\ldots,(-1)^{r}a_{1}^{-2r-1}\Delta_{2r+1}(p))=\\ \\ =1+\operatorname{{\rm V}}(a_{1},a_{1}^{-2}\Delta_{3}(p),a_{1}^{-4}\Delta_{5}(p),\ldots,a_{1}^{-2r}\Delta_{2r+1}(p))+l-r=\\ \\ =\operatorname{{\rm V}}(1,\Delta_{1}(p))+\operatorname{{\rm V}}(a_{1},a_{1}^{-2}\Delta_{3}(p),a_{1}^{-4}\Delta_{5}(p),\ldots,a_{1}^{-2r}\Delta_{2r+1}(p))+l-r=\\ \\ =\operatorname{{\rm V}}(1,\Delta_{1}(p),\Delta_{3}(p),\Delta_{5}(p),\ldots,\Delta_{2r+1}(p))+l-r,\end{array}

(5.28)

since $a_{1}=\Delta_{1}(p)<0$ . Now from (5.28) we obtain (5.25) for $r=l$ and (5.26) for $r=l-1$ .

Finally, let $n=2l+1$ and $a_{1}=\Delta_{1}(p)=0$ . The formulæ (5.22) and (2.21)–(2.22) yield

\begin{array}[]{c}k=l-\operatorname{{\rm V}}(1,\widehat{D}_{1}(\Phi),\widehat{D}_{2}(\Phi),\ldots,\widehat{D}_{r}(\Phi))=\\ \\ =l-\operatorname{{\rm V}}\left(1,-\dfrac{\Delta_{5}(p)}{\Delta_{3}(p)},\dfrac{\Delta_{7}(p)}{\Delta_{3}(p)},\ldots,(-1)^{r}\dfrac{\Delta_{2r+3}(p)}{\Delta_{3}(p)}\right)=\\ \\ =l-1-r+\operatorname{{\rm V}}(1,0,-a_{0}a_{3}^{2})+\operatorname{{\rm V}}\left(1,\dfrac{\Delta_{5}(p)}{\Delta_{3}(p)},\dfrac{\Delta_{7}(p)}{\Delta_{3}(p)},\ldots,\dfrac{\Delta_{2r+3}(p)}{\Delta_{3}(p)}\right)=\\ \\ =l-1-r+\operatorname{{\rm V}}(1,\Delta_{1}(p),\Delta_{3}(p))+\operatorname{{\rm V}}\left(\Delta_{3}(p),\Delta_{5}(p),\Delta_{7}(p),\ldots,\Delta_{2r+3}(p)\right)=\\ \\ =l-1-r+\operatorname{{\rm V}}(1,\Delta_{1}(p)\Delta_{3}(p),\Delta_{5}(p),\Delta_{7}(p),\ldots,\Delta_{2r+3}(p)),\end{array}

(5.29)

where $r=l-1$ for $p(0)\neq 0$ , and $r=l-2$ for $p(0)=0$ . Now (5.29) is equivalent to (5.25) for $r=l-1$ and to (5.26) for $r=l-2$ . ∎

Remark 5.7.

Using the formulæ (2.14)–(2.15) and (2.21)–(2.22), one can easily reformulate Theorem 5.6 in terms of the minors $\eta_{j}(p)$ defined in Definition 2.2.

Now from Theorems 1.28 and 5.3 we obtain the following theorem, which is a generalization of the famous Liénard and Chipart criterion of stability for the class of generalized Hurwitz polynomials.

Theorem 5.8 (Generalized Liénard and Chipart theorem).

The polynomial $p$ given in (5.1) is generalized Hurwitz if and only if the inequalities (5.24) hold. The order $k$ of the polynomial $p$ equals

k=v(a_{n},a_{n-2},\ldots,1)=v(a_{n},a_{n-1},a_{n-3},\ldots,1)\qquad\text{if}\qquad a_{n}\neq 0,

(5.30)

or²²²²22If $a_{n}=0$ , then always $a_{n-1}\neq 0$ , since the polynomials $p_{1}$ and $p_{0}$ , the numerator and the denominator of the R-function $\Phi$ , have no common roots.

k=v(a_{n-2},a_{n-4},\ldots,1)+1=v(a_{n-1},a_{n-3},\ldots,1)+1,\qquad\text{if}\qquad a_{n}=0,

(5.31)

where $v$ is the number of strong sign changes (see Definition 1.27).

Proof.

The fact that the polynomial $p$ is generalized Hurwitz if and only if the inequalities (5.24) hold follow from Theorem 5.6. Now we prove the formulæ (5.30)–(5.31).

Let $n=2l$ . By Theorem 5.3, $p$ is a generalized Hurwitz polynomial if and only if the function

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=\dfrac{a_{1}u^{l-1}+a_{3}u^{l-2}+\ldots+a_{2l-3}u+a_{2l-1}}{a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{2l-2}u+a_{2l}}

is an R-function. The number of its nonnegative poles is equal to order $k$ of the polynomial $p$ . Let $\Phi$ has exactly $m$ positive poles²³²³23Clearly, $m=k$ if $a_{n}\neq 0$ , and $m=k-1$ if $a_{n}=0$ .. We prove that

m=v(a_{n},a_{n-2},\ldots,1).

(5.32)

It will be equivalent to to the first equalities in (5.30)–(5.31), because $v$ is the number of strong sign changes, so $v(a_{n},a_{n-2},\ldots,1)=v(a_{n-2},a_{n-4},\ldots,1)$ if $a_{n}=0$ . Now Theorem 1.28 implies

m=v(a_{2l},a_{2l-2},\ldots,a_{0})=v(a_{2l},a_{2l-2},\ldots,a_{0})+v(a_{0},1)=v(a_{2l},a_{2l-2},\ldots,a_{0},1),

which is exactly (5.32).

Let us now consider the function

\Psi(u)=-\dfrac{1}{\Phi(u)}=-\dfrac{p_{0}(u)}{p_{1}(u)}=-\dfrac{a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{2l-2}u+a_{2l}}{a_{1}u^{l-1}+a_{3}u^{l-2}+\ldots+a_{2l-3}u+a_{2l-1}}.

This function is also an R-function by Theorem 1.23, since $\Phi$ is an R-function with exactly $k$ nonnegative poles, say $0\leqslant\omega_{1}<\ldots,\omega_{k}$ . Note that $\Phi(u)>0$ for $u\in(\omega_{k},+\infty)$ , so it can have at least $k-1$ and at most $k$ positive zeroes. There can be only four possibilities:

$\Phi(0)=\dfrac{a_{2l-1}}{a_{2l}}<0$ . Then $v(a_{2l},a_{2l-1})=1$ , and $\Phi$ has no zeroes on the interval $(0,\omega_{1})$ , since it is decreasing on this interval and $\Phi(u)\to-\infty$ as $u\nearrow\omega_{1}$ . Therefore, $\Phi$ has exactly $k-1$ positive zeroes, so $\Psi$ has exactly $k-1$ positive poles. By Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we have

\begin{array}[]{c}k-1=v(a_{2l-1},a_{2l-3},\ldots,a_{1})=-1+v(a_{2l},a_{2l-1})+v(a_{2l-1},a_{2l-3},\ldots,a_{1},1)+v(a_{1},1)=\\ \\ =-1+v(a_{2l},a_{2l-1},a_{2l-3},\ldots,a_{1},1),\end{array}

which is exactly

k=v(a_{n},a_{n-1},a_{n-3},\ldots,1).

(5.33)

Here we used the inequality $\Delta_{1}(p)=a_{1}>0$ , which follows from the inequalities (5.24) for $n=2l$ .

II.

$\Phi(0)=\dfrac{a_{2l-1}}{a_{2l}}>0$ . Then $v(a_{2l},a_{2l-1})=0$ , and $\Phi$ has exactly $k$ positive zeroes, so $\Psi$ has exactly $k$ positive poles. Thus, from Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we obtain

\begin{array}[]{c}k=v(a_{2l-1},a_{2l-3},\ldots,a_{1})=v(a_{2l},a_{2l-1})+v(a_{2l-1},a_{2l-3},\ldots,a_{1})+v(a_{1},1)=\\ \\ =v(a_{2l},a_{2l-1},a_{2l-3},\ldots,a_{1},1),\end{array}

which is exactly (5.33).

III.

$\Phi(0)=0$ . Then $\Psi$ has a pole at zero, that is, $a_{2l-1}=0$ . Since $\Psi$ is an R-function, it has positive residues at each its pole, so $\lim\limits_{u\to 0}u\Psi(u)=-\dfrac{a_{2l}}{a_{2l-3}}>0$ . Thus, $v(a_{2l},a_{2l-3})=v(a_{2l},a_{2l-1},a_{2l-3})=1$ . As in Case I, $\Phi$ has exactly $k-1$ positive zeroes, so the function $\Psi$ has exactly $k-1$ positive poles. Therefore, by Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we have

\begin{array}[]{c}k-1=v(a_{2l-3},a_{2l-5},\ldots,a_{1})=-1+v(a_{2l},a_{2l-1},a_{2l-3})+v(a_{2l-3},a_{2l-5},\ldots,a_{1})+v(a_{1},1)=\\ \\ =-1+v(a_{2l},a_{2l-1},a_{2l-3},\ldots,a_{1},1),\end{array}

which is precisely (5.33).

IV.

$\Psi(0)=0$ . Then $p(0)=a_{n}=a_{2l}=0$ , and the function $\Phi$ has a pole at zero, so it has exactly $k-1$ positive zeroes. Thus, the function $\Psi$ has exactly $k-1$ positive poles, and by Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we have

$\begin{array}[]{c}k-1=v(a_{2l-1},a_{2l-3},\ldots,a_{1})=v(a_{2l-1},a_{2l-3},\ldots,a_{1},1),\end{array}$

therefore,

$k=v(a_{n-1},a_{n-3},\ldots,1)+1.$ (5.34)

Let now $n=2l+1$ . By Theorem 5.3, $p$ is a generalized Hurwitz polynomial if and only if $\Phi$ is an R-function with exactly $l$ zeroes and $l$ or $l-1$ poles. If $k$ is order of the polynomial $p$ , then $\Phi$ has exactly $k$ nonnegative poles for $a_{1}>0$ and exactly $k-1$ nonnegative poles for $a_{1}\leqslant 0$ .

At first, we establish the formula (5.32), where $m$ is the number of positive poles of the function $\Phi$ . As we mentioned above, this formula is equivalent to the first equalities in (5.30)–(5.31).

If $a_{1}>0$ , then from Theorem 1.28 applied to the function $\Phi$ we obtain

m=v(a_{2l+1},a_{2l-1},\ldots,a_{1})=v(a_{2l+1},a_{2l-1},\ldots,a_{1},1).

This is exactly (5.32).

Let $a_{1}\leqslant 0$ . As we established above, if $a_{1}=0$ , then $a_{3}<0$ . Therefore, for $a_{1}\leqslant 0$ , we always have $v(a_{3},a_{1})=v(a_{3},a_{1},1)-1$ . Thus, Theorem 1.28 implies

m=v(a_{2l+1},a_{2l-1},\ldots,a_{1})=v(a_{2l+1},a_{2l-1},\ldots,a_{1},1)-1,

which is again (5.32), since for $a_{1}\leqslant 0$ , $m=k-1$ if $p(0)\neq 0$ , and $m=k-2$ if $p(0)=0$ .

As above, consider the function

\Psi(u)=-\dfrac{1}{\Phi(u)}=-\dfrac{p_{0}(u)}{p_{1}(u)}=-\dfrac{a_{1}u^{l}+a_{3}u^{l-1}+\ldots+a_{2l-1}u+a_{2l+1}}{a_{0}u^{l}+a_{2}u^{l-1}+\ldots+a_{2l-2}u+a_{2l}}.

This function is also an R-function by Theorem 1.23. As above, there can be only four possibilities:

$\Phi(0)=\dfrac{a_{2l}}{a_{2l+1}}<0$ , so $v(a_{2l+1},a_{2l})=1$ . If $a_{1}>0$ , then by Theorem 5.3, $\Phi$ has exactly $k$ positive poles, say $0<\omega_{1}<\ldots<\omega_{k}$ . Since $\Phi(u)\to\dfrac{a_{0}}{a_{1}}>0$ as $u\to+\infty$ and $\Phi$ is decreasing between its poles, the function $\Phi$ has no zeroes on the interval $(\omega_{k},+\infty)$ . Also $\Phi$ has no zeroes on the interval $(0,\omega_{1})$ , since $\Phi(0)<0$ by assumption. Thus, $\Phi$ has exactly $k-1$ positive zeroes, so $\Psi$ has exactly $k-1$ positive poles.

If $a_{1}\leqslant 0$ , then by Theorem 5.3, the function $\Phi$ has exactly $k-1$ positive poles, say $0<\omega_{1}<\ldots<\omega_{k-1}$ . Since $\Phi(u)$ is decreasing between its poles, it is negative for sufficiently large positive $u$ . Moreover, $\Phi(u)\nearrow+\infty$ as $u\searrow\omega_{k-1}$ . Thus, $\Phi$ has a zero on the interval $(\omega_{k-1},+\infty)$ , but it has no zeroes $(0,\omega_{1})$ , since $\Phi(0)<0$ by assumption. Consequently, $\Phi$ has exactly $k-1$ positive zeroes, so $\Psi$ has exactly $k-1$ positive poles.

Thus, we showed that the function $\Psi$ has exactly $k-1$ positive poles regardless of the sign of $a_{1}$ . Now from Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we obtain

\begin{array}[]{c}k-1=v(a_{2l},a_{2l-2},\ldots,a_{0})=-1+v(a_{2l+1},a_{2l})+v(a_{2l},a_{2l-2},\ldots,a_{0},1)=\\ \\ =-1+v(a_{2l+1},a_{2l},a_{2l-2},\ldots,a_{0},1),\end{array}

which is exactly (5.33).

II.

$\Phi(0)=\dfrac{a_{2l}}{a_{2l+1}}>0$ , so $v(a_{2l+1},a_{2l})=0$ . In the same way as above, one can show that the function $\Psi$ has exactly $k$ positive poles regardless of the sign of $a_{1}$ . Thus, from Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we obtain

\begin{array}[]{c}k=v(a_{2l},a_{2l-2},\ldots,a_{0})=v(a_{2l+1},a_{2l})+v(a_{2l},a_{2l-2},\ldots,a_{0},1)=\\ \\ =v(a_{2l+1},a_{2l},a_{2l-2},\ldots,a_{0},1).\end{array}

So the formula (5.33) is also valid in this case.

III.

$\Phi(0)=0$ . Then $\Psi$ has a pole at zero, that is, $a_{2l}=0$ . Since $\Psi$ is an R-function, it has positive residues at each its pole, so $\lim\limits_{u\to 0}u\Psi(u)=-\dfrac{a_{2l+1}}{a_{2l-2}}>0$ . Thus, $v(a_{2l+1},a_{2l-2})=v(a_{2l+1},a_{2l},a_{2l-2})=1$ . As above, one can show that $\Psi$ has exactly $k-1$ positive poles regardless of the sign of $a_{1}$ . Therefore, by Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ , we have

\begin{array}[]{c}k-1=v(a_{2l-2},a_{2l-4},\ldots,a_{0})=-1+v(a_{2l+1},a_{2l},a_{2l-2})+v(a_{2l-2},a_{2l-4},\ldots,a_{0},1)=\\ \\ =-1+v(a_{2l+1},a_{2l},a_{2l-2},\ldots,a_{0},1),\end{array}

which is precisely (5.33).

IV.

$\Psi(0)=0$ . Then $p(0)=a_{n}=a_{2l+1}=0$ , and the function $\Phi$ has a pole at zero. In the same way as above, one can prove that the function $\Psi$ has exactly $k-1$ positive poles regardless of the sign of $a_{1}$ , so Theorem 1.28 and Remark 1.29 applied to the function $\Psi$ imply

$\begin{array}[]{c}k-1=v(a_{2l},a_{2l-2},\ldots,a_{0}),\end{array}$

which is equivalent to (5.34), since $a_{0}>0$ .

∎

This theorem implies a necessary condition for polynomials to be generalized Hurwitz. This condition plays a role of Stodola’s necessary condition for Hurwitz stable polynomials (Theorem 3.2).

Corollary 5.9.

If the polynomial

p(z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n},\qquad a_{1},\dots,a_{n}\in\mathbb{R},\ a_{0}>0,

(5.35)

is generalized Hurwitz (of type I) of order $k$ , then

k=v(a_{n},a_{n-2},\ldots,1)=v(a_{n},a_{n-1},a_{n-3},\ldots,1)\qquad\text{if}\quad p(0)\neq 0,

k-1=v(a_{n},a_{n-2},\ldots,1)=v(a_{n},a_{n-1},a_{n-3},\ldots,1)\qquad\text{if}\quad p(0)=0,

This corollary implies Theorem 3.2 (Stodola’s theorem) for $k=0$ , and it implies Theorem 4.10 for $k=\left[\dfrac{n+1}{2}\right]$ and $a_{n}\neq 0$ .

Note that G. F. Korsakov [21] made an attempt to prove Theorem 5.8. However, his methods did not allow him to prove the whole theorem. He proved only sufficiency, that is, he proved that if the inequalities (5.24) hold for the polynomial (5.35), then $p$ is generalized Hurwitz, and the number $k$ of its positive zeroes can be found by the formulæ

•

for $n=2l$ ,

$k=v(a_{n},a_{n-2},\ldots,a_{2},a_{0});$ (5.36)
•

for $n=2l+1$ ,

$k=v(a_{n},a_{n-2},\ldots,a_{3},a_{1},a_{0}).$ (5.37)

Since in [21], it was assumed that $a_{n}\neq 0$ , the number $k$ in (5.36)–(5.37) is exactly order of the polynomial $p$ . Nevertheless, the generalized Hurwitz polynomials with a root at zero are also partially described in [21]. In fact, there was proved that if the following inequalities hold for the polynomial $p$ defined in (5.35)

\Delta_{n}(p)>0,\ \Delta_{n-2}(p)>0,\ \Delta_{n-4}(p)>0,\ldots,

(5.38)

then $p$ has exactly $k$ (defined by (5.36)–(5.37)) zeroes on the positive real half-axis, and other its zeroes are located in the open left half-plane. Generally speaking, the polynomial $p$ satisfied the inequalities (5.38) is not a generalized Hurwitz polynomial. However, if $p$ satisfies (5.38), then the polynomial $q(z)=zp(z)$ of degree $n+1$ is a generalized Hurwitz polynomial of order $k+1$ . Indeed, it is easy to see that the following equalities are true:

\Delta_{i}(q)=\Delta_{i}(p),\qquad i=1,\ldots,n.

So the inequalities (5.38) for the polynomial $p$ of degree $n$ are exactly the inequalities (5.24) hold for the polynomial $q$ of degree $n+1$ , so $q$ is a generalized Hurwitz polynomial having $k+1$ nonnegative zeroes.

At last, we establish a relation between generalized Hurwitz polynomials and continued fractions of Stieltjes type (if any).

Theorem 5.10.

If the polynomial $p$ of degree $n$ defined in (5.35) is generalized Hurwitz (of type I) of order $k$ and $\Delta_{n-2j}(p)\neq 0$ , $i=1,\ldots,l$ , where $\Delta_{0}(p)\equiv 1$ , then the function $\Phi$ has the following Stieltjes type continued fraction expansion:

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=c_{0}+\dfrac{1}{c_{1}u+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}u+\cfrac{1}{\ddots+\cfrac{1}{T}}}}},\qquad T=\begin{cases}&c_{2l}\qquad\text{if}\;\;\;p(0)\neq 0,\\ &c_{2l-1}u\quad\text{if}\;\;\;p(0)=0,\end{cases}

(5.39)

where $c_{0}=0$ if $n$ is even, and $c_{0}\neq 0$ if $n$ is odd, and

c_{2j-1}>0,\qquad j=1,\ldots,l.

(5.40)

Moreover, if $p(0)\neq 0$ , then the number of negative coefficients $c_{2j}$ , $j=0,1,\ldots,l$ , equals $k$ , and if $p(0)=0$ , then the number of negative coefficients $c_{2j}$ , $j=0,1,\ldots,l-1$ , equals $k-1$ .

Conversely, if for given polynomial $p$ of degree $n$ , its associated function $\Phi$ has a Stieltjes continued fraction (5.39)–(5.40), then $p$ is a generalized Hurwitz polynomial (with $\Delta_{n-2j}(p)\neq 0$ , $i=1,\ldots,l$ ), where $l$ defined in (2.2). Its order equals the number of negative coefficients $c_{2j}$ , $j=0,1,\ldots,l$ , if $p(0)\neq 0$ , or the number of negative coefficients $c_{2j}$ , $j=0,1,\ldots,l-1$ , plus one if $p(0)=0$ .

Proof.

The theorem follows from Theorem 5.6 and from the formulæ (3.13)–(3.14) ∎

Theorem 5.10 does not cover the case of $a_{1}=0$ . The following theorem fills this gap.

Theorem 5.11.

If the polynomial $p$ of degree $n=2l+1$ defined in (5.35) with $a_{1}=0$ is generalized Hurwitz (of type I) of order $k$ and $\Delta_{2j+1}(p)\neq 0$ , $i=1,\ldots,l-1$ , then the function $\Phi$ has the following Stieltjes continued fraction expansion:

\Phi(u)=\dfrac{p_{1}(u)}{p_{0}(u)}=-c_{-1}u+c_{0}+\dfrac{1}{c_{1}u+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}u+\cfrac{1}{\ddots+\cfrac{1}{T}}}}},\qquad T=\begin{cases}&c_{2l-2}\qquad\text{if}\;\;\;p(0)\neq 0,\\ &c_{2l-3}u\quad\text{if}\;\;\;p(0)=0,\end{cases}

(5.41)

where $c_{0}\in\mathbb{R}$ , and

c_{2j-1}>0,\qquad j=0,1,\ldots,l-1.

(5.42)

Moreover, if $p(0)\neq 0$ , then the number of negative coefficients $c_{2j}$ , $j=1,\ldots,l-1$ , equals $k-1$ . But if $p(0)=0$ , then the number of negative coefficients $c_{2j}$ , $j=0,1,\ldots,l-2$ , equals $k-2$ .

Conversely, if for given polynomial $p$ of degree $n=2l+1$ , its associated function $\Phi$ has a Stieltjes continued fraction (5.41)–(5.42), then $p$ is a generalized Hurwitz polynomial (with $\Delta_{2j+1}(p)\neq 0$ , $i=1,\ldots,l-1$ and $\Delta_{1}(p)=a_{1}=0$ ). Its order equals the number of negative coefficients $c_{2j}$ , $j=1,\ldots,l-1$ , plus one if $p(0)\neq 0$ , or the number of negative coefficients $c_{2j}$ , $j=1,\ldots,l-2$ , plus two if $p(0)=0$ .

Proof.

Let the polynomial $p$ of degree $n=2l+1$ with $a_{1}=0$ be generalized Hurwitz (of type I) of order $k$ and $\Delta_{2j+1}(p)\neq 0$ , $i=1,\ldots,l-1$ . By Theorem 5.5 and by formulæ (2.21), we obtain $a_{0}a_{3}<0$ and

\begin{cases}&D_{j}(\Phi)>0,\qquad j=1,\ldots,l-1,\\ &\widehat{D}_{j}(\Phi)\neq 0\qquad j=1,\ldots,l-2.\end{cases}

(5.43)

It is easy to see that for the function $F(u):=\Phi(u)+c_{-1}u$ , where $c_{-1}=-\dfrac{a_{0}}{a_{3}}$ , the following equalities hold

\begin{cases}&D_{j}(F)=D_{j}(\Phi),\\ &\widehat{D}_{j}(F)=\widehat{D}_{j}(\Phi)\neq 0,\end{cases}\qquad j=1,\ldots,l-1,

(5.44)

so the function $F$ has a Stiltjes type continued fraction (1.33) with $c_{2j-1}>0$ , $j=1,\ldots,l-1$ , that follows from (5.43)–(5.44) and (1.38). Also from (5.22), (5.43)–(5.44) and (1.37) we obtain that the number of negative coefficients $c_{2j}$ , $j=1,\ldots,l-1$ , equals $k-1$ if $p(0)\neq 0$ , or $k-2$ if $p(0)=0$ .

The converse assertion of the theorem follows from the formulæ (5.43)–(5.44), from Theorem 1.36 applied to the function $F(u):=\Phi(u)-\dfrac{a_{0}}{a_{3}}u$ , and from Theorem 5.5. ∎

Theorem 1.22 implies the following theorem.

Theorem 5.12.

Let the polynomial $p$ of degree $n\geqslant 2$ as in (5.35) be generalized Hurwitz polynomial. Then all polynomials

p_{j}(z)=\sum\limits_{i=0}^{n-2j}\left[\dfrac{n-i}{2}\right]\left(\left[\dfrac{n-i}{2}\right]-1\right)\cdots\left(\left[\dfrac{n-i}{2}\right]+j-1\right)a_{i}z^{n-2j-i},\quad k=1,\ldots,\left[\dfrac{n}{2}\right]-1,

are also generalized Hurwitz.

5.2 Application to bifurcation theory. Polynomials dependent on parameters.

Let us consider a system of ordinary differential equations dependent on a real parameter, say $\alpha$ :

\begin{cases}&\dot{x}(t)=F(x(t),\alpha)=A(\alpha)x(t)+o(x(t)),\\ &x(t_{0})=x_{0}\end{cases}\qquad F(x),\,\,x\in\mathbb{R}^{n},

(5.45)

where the $n\times n$ matrix $A(\alpha)$ is the linear part of the function $F(x,\alpha)$ , and $F(0,\alpha)\equiv 0$ . Suppose that for some value of the parameter $\alpha_{0}$ all the eigenvalues of the matrix $A(\alpha_{0})$ are located in the open left half-plane of the complex plane. In this case, the stability theory says [26, 28, 18] that the zero solution of the system (5.45) is (asymptotically) stable, that is, all the solutions of the system (5.45), whose initial conditions $x_{0}$ are sufficiently small, tend to zero as $t\to+\infty$ .

When the parameter $\alpha$ arrives at its critical value, say $\alpha_{1}^{*}$ the following two basic cases can occur, generically: either a pair of complex conjugate eigenvalues (of multiplicity one) of the matrix $A(\alpha_{1}^{*})$ appear on the imaginary axis or an eigenvalue (of multiplicity one) becomes $0$ . In the former case, for $\alpha=\alpha_{1}^{*}+\varepsilon$ , where $\varepsilon>0$ is sufficiently small, there appears a small-amplitude limit cycle branching from the zero solution (a fixed point) of the system (5.45), but the zero solution loses its stability²⁴²⁴24Here we assume that for $\alpha>\alpha_{1}^{*}$ , the matrix $A(\alpha)$ has an eigenvalue with a positive real part.. This type of bifurcation is called Hopf bifurcation [26, 28, 18]. In hydrodynamics [42], such type of bifurcation is called an oscillatory loss of stability or oscillatory instability. In the case, when the matrix $A(\alpha_{1}^{*})$ has all the eigenvalues in the open left half-plane except one, which is equal to zero and is of multiplicity one, new fixed points of the system (5.45) branch from the zero fixed point. In hydrodynamics [42], such type of bifurcation is called a monotone loss of stability or monotone instability. To determine the type of bifurcation of a fixed point of a given system is a very important problem in hydrodynamics and mechanics (see [28, 42] and references there).

Let us consider the characteristic polynomial $p(z,\alpha)=\det(zE-A(\alpha))$ of the matrix $A(\alpha)$ . By assumption $p(z,\alpha_{0})$ is Hurwitz stable. Recall that $\alpha_{1}^{*}>\alpha_{0}$ is the critical value of the parameter $\alpha$ such that the zero solution of the system (5.45) loses its stability. The monotonicity principle takes place if and only if the polynomial $p(z,\alpha_{1}^{*})$ is a generalized Hurwitz polynomial, that is, the inequalities (5.24) hold. This fact follows from Theorem 5.6. Moreover, the polynomial $p(z,\alpha)$ is generalized Hurwitz of order $0$ (Hurwitz stable) for $\alpha<\alpha_{1}*$ , and $p(z,\alpha_{1}^{*})$ is a generalized Hurwitz polynomial of order $1$ .

Now we describe further changes of zero location of the polynomial $p(z,\alpha)$ when $\alpha$ goes beyond $\alpha_{1}^{*}$ until $p(z,\alpha)$ becomes self-interlacing. Let $\{\alpha_{2}^{*},\alpha_{3}^{*},\ldots,\alpha_{m}^{*}\}$ , $2\leqslant m\leqslant\left[\dfrac{n+1}{2}\right]$ , be the next critical values of the parameter $\alpha$ at each of which order of the generalized Hurwitz polynomial $p(z,\alpha)$ increases. This means that $p(z,\alpha_{i}^{*})$ has a simple root at zero, and for $\alpha>\alpha_{i}^{*}$ sufficiently close to $\alpha_{i}^{*}$ , this root becomes a positive simple root of $p(z,\alpha)$ with the minimal absolute value (among real roots of $p(z,\alpha)$ ). Therefore, when $\alpha$ transfers $\alpha_{m}^{*}$ , the polynomial $p(z,\alpha)$ has exactly $m$ positive simple zeroes for $\alpha>\alpha_{m}^{*}$ sufficiently close to $\alpha_{m}^{*}$ . Denote the minimal positive zero of $p(z,\alpha)$ by $\mu_{m}$ . Obviously, if the difference $\alpha-\alpha_{m}^{*}>0$ is sufficiently small, then the number $\mu_{m}>0$ is close to zero, so $p(z,\alpha)$ has no zeroes on the interval $(-\mu_{m},0)$ . Let $\alpha$ increase. One more simple positive zero of the polynomial $p(z,\alpha)$ can appear²⁵²⁵25It is possible only if $m<\left[\dfrac{n+1}{2}\right]$ ., when two zeroes of a pair of complex conjugate zeroes of $p(z,\alpha)$ (all its nonreal zeroes lie in the open left half-plane) face each other on the interval $(-\mu_{m},0)$ to become a real zero of $p(z,\alpha)$ of multiplicity two. Further, when $\alpha$ grows, this zero splits into to real simple roots on the interval $(-\mu_{m},0)$ . One of those roots goes toward zero and crosses it, when $\alpha$ transfers through the next critical value $\alpha_{m+1}^{*}$ of the parameter $\alpha$ . Thus, for sufficiently small difference $\alpha-\alpha_{m+1}^{*}>0$ , this zero becomes the smallest positive zero $\mu_{m+1}$ . The second zero of the mentioned pair of zeroes remains on the interval $(-\mu_{m},-\mu_{m+1})$ , since it cannot cross the value $-\mu_{m}$ and cannot go away from the real axis.

Let us show that there are no any other scenarios for positive zeroes to appear if $p(z,\alpha)$ is generalized Hurwitz for all $\alpha$ . In fact, suppose that for $\alpha>\alpha_{m}^{*}$ , some negative zeroes of $p(z,\alpha)$ come into the interval $(-\mu_{m},0)$ from the real axis, that is, from the interval $(-\infty,-\mu_{m})$ . But in this case, the polynomial $p(z,\alpha)$ will have zeroes $\pm\mu_{m}$ at some value of $\alpha$ , so by the Orlando’s formula [12, 16], we have $\Delta_{n+1}(p)=0$ . According to Theorem 5.6, this contradicts the assumption that $p(z,\alpha)$ is generalized Hurwitz.

Thus, when $\alpha$ increases from $\alpha_{0}$ , at each interval $(-\mu_{j},-\mu_{j+1})$ the polynomial $p(z,\alpha)$ has an odd number of zeroes. One of those zeroes appears on $(-\mu_{j},-\mu_{j+1})$ as it was described above, while the other zeroes appear from complex conjugate pairs. The maximal number of positive zeroes of $p(z,\alpha)$ equals $\left[\dfrac{n+1}{2}\right]$ . In this case, $p(z,\alpha)$ becomes self-interlacing.

So the monotonicity principle takes place for the system (5.45) if and only if for the first critical value $\alpha_{1}^{*}$ of the parameter $\alpha$ the characteristic polynomial $p(z,\alpha_{1}^{*})=\det(zE-A(\alpha_{1}^{*}))$ is generalized Hurwitz, that is, the inequalities (5.24) hold.

Acknowledgement

I am grateful to Yu.S. Barkovsky and O. Holtz for helpful discussions and for Barkovsky’s detailed suggestions for the improvement of some results in the paper.

References

[1] N. I. Akhiezer. The classical moment problem and some related questions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York, 1965.
[2] N. I. Akhiezer and M. G. Krein. Some questions in the theory of moments. translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2. American Mathematical Society, Providence, R.I., 1962.
[3] B. A. Asner, Jr. On the total nonnegativity of the Hurwitz matrix. SIAM J. Appl. Math., 18:407–414, 1970.
[4] F. V. Atkinson. Discrete and continuous boundary problems. Mathematics in Science and Engineering, Vol. 8. Academic Press, New York, 1964.
[5] Yu. S. Barkovsky. Private communication, 2004.
[6] Yu. S. Barkovsky. Lectures on the Routh-Hurwitz problem. arXiv:0802.1805, 2008.
[7] N. G. Čebotarev and N. N. Meĭman. The Routh-Hurwitz problem for polynomials and entire functions. Real quasipolynomials with $r=3$ , $s=1$ . Trudy Mat. Inst. Steklov., 26:331, 1949. Appendix by G. S. Barhin and A. N. Hovanskiĭ.
[8] S. Fisk. Polynomials, roots, and interlacing. arXiv:0612833, 2006.
[9] G. Frobenius. Über das Trägheitsgesetz der quadratischen Formen. Sitz.-Ber. Acad. Wiss. Phys.-Math. Klasse, Berlin, pages 241–256; 407–431, 1894.
[10] F. P. Gantmacher and M. G. Krein. Oscillation matrices and kernels and small vibrations of mechanical systems. AMS Chelsea Publishing, Providence, RI, revised edition, 2002. Translation based on the 1941 Russian original, Edited and with a preface by Alex Eremenko.
[11] F. R. Gantmacher. The theory of matrices. Vol. 1. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959.
[12] F. R. Gantmacher. The theory of matrices. Vol. 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959.
[13] C. Hermite. On the number of roots of an algebraic equation contained between given limits. Internat. J. Control, 26(2):183–195, 1977. Translated from the French original by P. C. Parks, Routh centenary issue.
[14] O. Holtz. Hermite-Biehler, Routh-Hurwitz, and total positivity. Linear Algebra Appl., 372:105–110, 2003.
[15] O. Holtz. The inverse eigenvalue problem for symmetric anti-bidiagonal matrices. Linear Algebra Appl., 408:268–274, 2005.
[16] O. Holtz and Tyaglov M. Structured matrices, continued fractions, and root localization of polynomials. arXiv:0912.4703, 2009.
[17] A. Hurwitz. Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. In Stability theory (Ascona, 1995), volume 121 of Internat. Ser. Numer. Math., pages 239–249. Birkhäuser, Basel, 1996. Reprinted from Math. Ann. 46 (1895), 273–284 [JFM 26.0119.03].
[18] G. Iooss and D. Joseph. Elementary stability and bifurcation theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990.
[19] I. V. Kac, , and M. G. Krein. On the spectral functions of the string, volume 103 of Amer. Math. Soc. Transl. American Mathematical Society, 1974.
[20] J. H. B. Kemperman. A Hurwitz matrix is totally positive. SIAM J. Math. Anal., 13(2):331–341, 1982.
[21] G. F. Korsakov. A generalization of a theorem of Liénard and Chipart. Mat. Zametki, 22(1):13–21, 1977.
[22] M. G. Kreĭn and M. A. Naĭmark. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra, 10(4):265–308, 1981. Translated from the Russian by O. Boshko and J. L. Howland.
[23] M. G. Kreĭn and A. A. Nudel^′man. The Markov moment problem and extremal problems. American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, Vol. 50.
[24] L. Kronecker. Algebraische Reduction der Schaaren bilinearer Formen. S.-B. Akad. Berlin, 1890.
[25] A. Liénard and M. Chipart. Sur la signe de la partie réelle des racines d’une équation algébraique.
[26] A. M. Lyapunov. The general problem of the stability of motion. Taylor & Francis Ltd., London, 1992. Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A. T. Fuller, With an introduction and preface by Fuller, a biography of Lyapunov by V. I. Smirnov, and a bibliography of Lyapunov’s works compiled by J. F. Barrett, Lyapunov centenary issue, Reprint of Internat. J. Control 55 (1992), no. 3 [ MR1154209 (93e:01035)], With a foreword by Ian Stewart.
[27] A. A. Markov. Collected works (Russian). Academy of Scines USSR, Moscow, 1948.
[28] J. E. Marsden and M. McCracken. The Hopf bifurcation and its applications. Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale, Applied Mathematical Sciences, Vol. 19.
[29] G. Pick. Über die Beschränkungen analytische Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden. Math. Ann., 77:7–23, 1916.
[30] V. Pivovarchik. On spectra of a certain class of quadratic operator pencils with one-dimensional linear part. Ukraïn. Mat. Zh., 59(5):702–716, 2007.
[31] V. Pivovarchik. Symmetric Hermite-Biehler polynomials with defect. In Operator theory in inner product spaces, volume 175 of Oper. Theory Adv. Appl., pages 211–224. Birkhäuser, Basel, 2007.
[32] V. Pivovarchik and C. van der Mee. The inverse generalized Regge problem. Inverse Problems, 17(6):1831–1845, 2001.
[33] V. Pivovarchik and H. Woracek. Shifted Hermite-Biehler functions and their applications. Integral Equations Operator Theory, 57(1):101–126, 2007.
[34] V. Pivovarchik and H. Woracek. The square-transform of Hermite-Biehler functions. A geometric approach. Methods Funct. Anal. Topology, 13(2):187–200, 2007.
[35] E. J. Routh. A treatise on the stability of a given state of motion. London, 1877.
[36] T. Sheil-Small. Complex polynomials, volume 75 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002.
[37] B. Simon. The classical moment problem as a self-adjoint finite difference operator. Adv. Math., 137(1):82–203, 1998.
[38] T. J. Stieltjes. Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Math. (6), 4(3):J76–J122, 1995. Reprint of Ann. Fac. Sci. Toulouse 8 (1894), J76–J122.
[39] T. J. Stieltjes. Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse Math. (6), 4(4):A5–A47, 1995. Reprint of Ann. Fac. Sci. Toulouse 9 (1895), A5–A47.
[40] C. van der Mee and V. Pivovarchik. The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter. Opuscula Math., 25(2):243–260, 2005.
[41] H. S. Wall. Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc., New York, 1948.
[42] V. I. Yudovich. Eleven great problems of mathematical hydrodynamics. Mosc. Math. J., 3(2):711–737, 746, 2003. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.