Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials $t_{n}(x,N)$ can be defined as a special case of the Hahn polynomials [1, p.174]

(1.1)

$$\begin{split}Q_{n}(x;\alpha,\beta,N)&={{}_{3}F_{2}}(-n,-x,n+\alpha+\beta+1;-N,\alpha+1;1)\\ &=\sum_{k=0}^{n}\frac{(-n)_{k}(-x)_{k}(n+\alpha+\beta+1)_{k}}{(-N)_{k}(\alpha+1)_{k}k!}.\end{split}$$

With $\alpha=\beta=0$ in (1.1), we have

(1.2)

$$\begin{split}t_{n}(x,N)&=(-1)^{n}(N-n)_{n}Q_{n}(x;0,0,N-1)\\ &=(-1)^{n}(N-n)_{n}\sum_{k=0}^{n}\frac{(-n)_{k}(-x)_{k}(n+1)_{k}}{(-N+1)_{k}k!k!};\end{split}$$

see [1, p.176].

In a recent paper [8], we have studied the asymptotic behaviour of $t_{n}(x,N+1)$ as $n$, $N\rightarrow\infty$ in such a way that the ratios

(1.3)

$$a=x/N\qquad\text{and}\qquad b=n/N,$$

satisfy the inequalities

(1.4)

$$-\infty<a\leq\frac{1}{2}\qquad\text{and}\qquad 0<b<1.$$

In view of the symmetry relation [8, eq.(8)]

(1.5)

$$t_{n}(x,N+1)=(-1)^{n}t_{n}(N-x,N+1),$$

our study actually covers the entire real-axis $-\infty<x<\infty$ or, equivalently, the entire parameter range $-\infty<a<\infty$. For a discussion of the asymptotic behaviour of the Hahn polynomials, see [4].

In the present paper, we shall investigate the behaviour of the polynomials $t_{n}(x,N+1)$, when the parameter $b$ given in (1.3) either tends to $0$ or tends to $1$. The case $b\rightarrow 1$ turns out to be rather simple, since the ultimate expansions and their derivations are similar to those in the case when $b$ is fixed; see (2.21) for $0\leq a\leq\frac{1}{2}$ and (2.28) for $a<0$.

To derive asymptotic approximation for $t_{n}(x,N+1)$ when $b\rightarrow 0$, we divide our discussion into several cases, depending on the quantities $n$, $x$ and $xN/n^{2}$. A summary of our findings is given in Table 1 below.

Some explanation is needed for this table. By “small”, “fixed” and “large”, we mean, respectively, $x=o(1)$, $x\in[\delta,M]$ and $x\gg O(1)$ as $N\rightarrow\infty$, where $\delta$ is a small positive number and $M$ is a large positive number. By “Airy”, we mean an asymptotic expansion whose associated approximants are the Airy functions $\text{Ai}(z)$ and $\text{Bi}(z)$. In the same manner, by “Bessel” and “Kummer”, we mean asymptotic expansions whose associated approximants are, respectively, the Bessel function $J_{v}(z)$ and the Kummer function $M(a,c,z)$. The Kummer function used in this paper is defined by

(1.6)

$$\mathbf{M}(a,c,z)=\frac{1}{\Gamma(c)}M(a,c,z)=\frac{1}{\Gamma(c)}\sum_{s=0}^{\infty}\frac{(a)_{s}}{(c)_{s}}\frac{z^{s}}{s!},$$

which was introduced in [6, p.255]. The case involving Bessel functions covers an earlier result of Sharapodinov [10], where, instead of Bessel functions, Jacobi polynomials are used as an approximant. In the case when $xN/n^{2}$ is large and $x$ is either fixed or small, the series in (1.2) is itself an asymptotic expansion (in the generalized sense [11, p.10]) as $N\rightarrow\infty$. Hence, there is no need to seek for another asymptotic representation.

The arrangement of the present paper is as follows. In Section 2, we recall the major results in [8]; to facilitate our presentation for later sections, we also include here brief derivations of the expansions given in [8]. In Section 3, we consider the case when $xN/n^{2}$ is small and show that the expansions are of Kummer-type. In Section 4, we consider the case when the quantity $xN/n^{2}$ is fixed; that is, when $xN/n^{2}$ is bounded away from zero and infinity. There are three subcases, depending on $x$ being large, fixed or small. In all three subcases, we will show that the expansions are of Airy-type; see Table 1. The case when both quantities $x$ and $xN/n^{2}$ are large is dealt with in Section 5, where we show that the expansion of $t_{n}(x,N+1)$ can be expressed in terms of Bessel functions. The final section is devoted to two remaining cases, namely, (i) when $xN/n^{2}$ is large and $x$ is either fixed or small, and (ii) $x<0$.

In [8], we have divided our discussion into two cases: (i) $0\leq a\leq\frac{1}{2}$, and (ii) $a<0$. As mentioned in Section 1, by virtue of the symmetry relation (1.5), these two cases cover the entire range $-\infty<a<\infty$.

In case (i), we started with the integral representation

(2.1)

$$t_{n}(x,N+1)=\frac{(-1)^{n}}{2\pi i}\frac{\Gamma(n+N+2)}{\Gamma(n+1)\Gamma(N-n+1)}\int_{0}^{1}\int_{\gamma_{1}}\frac{1}{w-1}e^{Nf(t,w)}\mathrm{d}w\mathrm{d}t,$$

where

(2.2)

$$f(t,w)=b\ln(1-t)+(1-b)\ln t+a\ln w-a\ln(w-1)+b\ln[1-(1-t)w],$$

and the curve $\gamma_{1}$ starts at $w=0$, runs along the lower edge of the positive real line towards $w=1$, encircles the point $w=1$ in the counterclockwise direction and returns to the origin along the upper edge of the positive real line. As a function of two variables, the partial derivatives $\partial f/\partial t$ and $\partial f/\partial w$ vanish at $(t,w)=(t_{\pm},w_{\pm})$, where

(2.3)

$$t_{+}=\frac{2-2a-b^{2}+b\sqrt{b^{2}-4a+4a^{2}}}{2(1-a)(1+b)},\qquad w_{+}=\frac{b+\sqrt{b^{2}-4a+4a^{2}}}{2b},$$

and

(2.4)

$$t_{-}=\frac{2-2a-b^{2}-b\sqrt{b^{2}-4a+4a^{2}}}{2(1-a)(1+b)},\qquad w_{-}=\frac{b-\sqrt{b^{2}-4a+4a^{2}}}{2b}.$$

For each fixed $w\in\gamma_{1}$, we now find a steepest descent path of the phase function $f(t,w)$ in the variable $t$, which passes through a saddle point $t_{0}(w)$ depending on $w$. (Note that not only the saddle point $t_{0}(w)$, but all points $t$ on this steepest descent path depend on $w$.) The function $f(t_{0}(w),w)$ is a function of $w$ alone. To find the relevant saddle point $t_{0}(w)$, we solve the equation $\partial f(t,w)/\partial t=0$, and obtain

(2.5)

$$t_{0}(w)=\frac{2w-1+\sqrt{1+4b^{2}(w-1)w}}{2(1+b)w}.$$

It can be shown that

(2.6)

$$t_{0}(w_{+})=t_{+},\qquad t_{0}(w_{-})=t_{-}.$$

Define the standard transformation $t\rightarrow\tau$ by

(2.7)

$$\tau^{2}=f(t_{0}(w),w)-f(t,w);$$

see [11, p.88]. Note that we have $\tau=-\infty$ when $t=0$, and $\tau=+\infty$ when $t=1$. Furthermore, this mapping takes $t=t_{0}(w)$ to $\tau=0$. Coupling $(\ref{IR of DCP for x>0})$ and $(\ref{mapping t to tau})$ gives

(2.8)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n}}{2\pi i}\frac{\Gamma(n+N+2)}{\Gamma(n+1)\Gamma(N-n+1)}\\ &\times\int_{-\infty}^{+\infty}\int_{\gamma_{1}}\frac{1}{w-1}e^{Nf({t_{0}(w)},w)}e^{-N\tau^{2}}\frac{\mathrm{d}t}{\mathrm{d}\tau}\mathrm{d}w\mathrm{d}\tau,\end{split}$$

where $\gamma_{1}$ is the integration path of the variable $w$ in (2.1). Note that the first variable of $f$ in (2.8) is now $t_{0}(w)$, instead of $t$. Hence, the phase function is a function of $w$ alone. Setting $\partial f(t_{0}(w),w)/\partial w=0$, we obtain the saddle points

(2.9)

$$w_{\pm}=\frac{b\pm\sqrt{b^{2}-4a+4a^{2}}}{2b};$$

cf.(2.3) and (2.4). Motivated by (2.2), define the new phase function

(2.10)

$$\psi(u)=a\ln u-a\ln(u-1)+\eta u.$$

The saddle points of $\psi$ are given by

(2.11)

$$u_{\pm}=\frac{\eta\pm\sqrt{\eta^{2}+4a\eta}}{2\eta}.$$

To reduce the double integral in (2.1) into a canonical form, we define the second mapping $w\rightarrow u$ by

(2.12)

$$\begin{split}f(t_{0}(w),w)&=\psi(u)+\gamma\\ &=a\ln u-a\ln(u-1)+\eta u+\gamma\end{split}$$

with

(2.13)

$$u(w_{+})=u_{-},\quad\quad\quad\quad u(w_{-})=u_{+},$$

where $\eta$ and $\gamma$ are real numbers depending on the parameters $a$ and $b$ in (2.2). From (2.12), we have

(2.14)

$$\begin{split}\frac{\mathrm{d}w}{\mathrm{d}u}&=\frac{\mathrm{d}\psi}{\mathrm{d}u}\left/\frac{\mathrm{d}f(t_{0}(w),w)}{\mathrm{d}w}\right.\\ &=\frac{\eta(u-u_{+})(u-u_{-})w(1-w)\left[(2a-1)-\sqrt{1+4b^{2}w^{2}-4b^{2}w}\right]}{-2b^{2}u(u-1)(w-w_{+})(w-w_{-})}\end{split}$$

for $u\neq u_{\pm}$, and by L’H$\hat{o}$pital’s rule,

(2.15)

$$\frac{\mathrm{d}w}{\mathrm{d}u}=\left[\frac{\sqrt{\eta^{2}+4a\eta}(1-2a)(1-a)(-\eta)}{b^{3}\sqrt{b^{2}-4a+4a^{2}}}\right]^{1/2}$$

for $u=u_{\pm}$. With the change of variable $w\rightarrow u$ defined in (2.12), the representation in (2.8) becomes

(2.16)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n}}{2\pi i}\frac{\Gamma(n+N+2)}{\Gamma(n+1)\Gamma(N-n+1)}e^{N\gamma}\\ &\qquad\times\int_{-\infty}^{+\infty}\int_{\gamma_{u}}\frac{h(u,\tau)}{u-1}e^{N\psi(u)-N\tau^{2}}\mathrm{d}u\mathrm{d}\tau,\end{split}$$

where

(2.17)

$$h(u,\tau)=\frac{u-1}{w-1}\frac{\mathrm{d}w}{\mathrm{d}u}\frac{\mathrm{d}t}{\mathrm{d}\tau},$$

and $\gamma_{u}$ is the steepest descent path of $\psi(u)$ in the $u$-plane. An asymptotic expansion, holding uniformly for $a\in[0,\frac{1}{2}]$, was then derived in [8] by an integration-by-parts technique. To state the result, we define recursively $h_{0}(u,\tau)=h(u,\tau)$,

(2.18)

$$h_{l}(u,\tau)=a_{l}(\tau)+b_{l}(\tau)u-(u-u_{-})(u-u_{+})g_{l}(u,\tau)$$

and

(2.19)

$$h_{l+1}(u,\tau)=(u-1)\left[g_{l}(u,\tau)+u\frac{\partial g_{l}(u,\tau)}{\partial u}\right],\qquad l=0,1,2,....$$

Furthermore, we write

(2.20)

$$a_{l}(\tau)=\sum_{j=0}^{\infty}a_{l,j}\tau^{j},\qquad b_{l}(\tau)=\sum_{j=0}^{\infty}b_{l,j}\tau^{j}.$$

The resulting expansion takes the form

(2.21)

$$\begin{split}t_{n}(x,N+1)\sim\frac{(-1)^{n}\Gamma(n+N+2)e^{N\gamma}}{\Gamma(n+1)\Gamma(N-n+1)\sqrt{N}}&\left\{\mathbf{M}(aN+1,1,\eta N)\sum_{l=0}^{\infty}\frac{c_{l}}{(\eta N)^{l}}\right.\\ &\left.\text{ }+\mathbf{M}^{\prime}(aN+1,1,\eta N)\sum_{l=0}^{\infty}\frac{d_{l}}{(\eta N)^{l}}\right\},\end{split}$$

where $\mathbf{M}(a,c,z)$ is the Kummer function defined in (1.6), and the coefficients $c_{l}$ and $d_{l}$ are given explicitly by

(2.22)

$$c_{l}=\sum_{m=0}^{l}a_{l-m,2m}\Gamma(m+\frac{1}{2})\eta^{m},\qquad d_{l}=\sum_{m=0}^{l}b_{l-m,2m}\Gamma(m+\frac{1}{2})\eta^{m}.$$

Case (ii) was dealt with in a similar manner. We started with the double-integral representation

(2.23)

$$t_{n}(x,N+1)=\frac{(-1)^{n}}{2\pi i}\frac{\Gamma(n+N+2)}{\Gamma(n+1)\Gamma(N-n+1)}\int_{0}^{1}\int_{\gamma_{2}}\frac{1}{w-1}e^{N\widetilde{f}(t,w)}\mathrm{d}w\mathrm{d}t,$$

where

(2.24)

$$\widetilde{f}(t,w)=b\ln(1-t)+(1-b)\ln t+a\ln(-w)-a\ln(1-w)+b\ln[1-(1-t)w],$$

and the integration path $\gamma_{2}$ starts at $w=1$, traverses along the upper edge of the positive real line towards $w=0$, encircles the origin in the counterclockwise direction and returns to $w=1$ along the lower edge of the positive real line.

Now we follow the same argument as given in Case (i), and use the same mapping $t\rightarrow\tau$ defined in (2.7), except with $f(t,w)$ replaced by $\widetilde{f}(t,w)$. The result is

(2.25)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n}}{2\pi i}\frac{\Gamma(n+N+2)}{\Gamma(n+1)\Gamma(N-n+1)}\\ &\times\int_{-\infty}^{+\infty}\int_{\gamma_{2}}\frac{1}{w-1}e^{N\widetilde{f}({t_{0}(w)},w)}e^{-N\tau^{2}}\frac{\mathrm{d}t}{\mathrm{d}\tau}\mathrm{d}w\mathrm{d}\tau.\end{split}$$

Because of the shape of the contour $\gamma_{2}$, it turns out that the phase function $\widetilde{f}(t_{0}(w),w)$ has only one relevant saddle point, namely $w_{-}$; see (2.9). As a consequence, we define the mapping $w\rightarrow u$ by

(2.26)

$$\widetilde{f}(t_{0}(w),w)=a\ln(-u)-u+\gamma$$

with

(2.27)

$$u(w_{-})=a,$$

where $\gamma$ is a constant depending on the parameters $a$ and $b$ in (1.3). The final expansion is in terms of the gamma function, and we have

(2.28)

$$t_{n}(x,N+1)\sim\frac{(-1)^{n}\Gamma(n+N+2)N^{-aN}e^{N\gamma}}{\Gamma(n+1)\Gamma(N-n+1)\Gamma(-aN+1)}\sum_{l=0}^{\infty}\frac{c_{l}}{N^{l+\frac{1}{2}}},$$

where $c_{l}$ are constants that can be given recursively.

As $b\rightarrow 0$, some of the steps in Section 2 are no longer valid. Let us first examine the mapping $t\rightarrow\tau$ given in (2.7). In this mapping, we have used the fact that for each fixed $w\in\gamma_{1}$, the saddle point $t=t_{0}(w)$ in (2.5) is bounded away from $t=0$ and $t=1$, and the steepest descent path from $t=0$ to $t=1$, passing through $t_{0}(w)$, is mapped onto $(-\infty,\infty)$ in the $\tau$-plane. However, from (2.5), we have for any fixed $w$

(3.1)

$$t_{0}(w)=1-b+O\left(b^{2}\right)\qquad\text{as }b\rightarrow 0;$$

that is, as $b\rightarrow 0$, $t_{0}(w)$ approaches the branch point $t=1$ in the t-plane. Thus, the mapping (2.7) is no longer suitable in this case. Since we are more interested in the neighbourhood of $t=1$, where the term $b\ln(1-t)$ in the phase function (2.2) becomes singular, we introduce the mapping

(3.2)

$$f(t,w)=b\ln\tau-\tau+A,$$

where the constant $A$ does not depend on $t$ or $\tau$ (but may depend on $w$); see (3.4) below. To make the mapping $t\rightarrow\tau$ defined in (3.2) one-to-one and analytic, we prescribe $t=t_{0}(w)$ to correspond to $\tau=b$, which is the saddle point of $b\ln\tau-\tau+A$; i.e.,

(3.3)

$$\tau(t_{0}(w))=b.$$

This gives

(3.4)

$$A=f(t_{0}(w),w)-b\ln b+b.$$

Note that we have $\tau=0$ when $t=1$, and $\tau=+\infty$ when $t=0$. Furthermore, from (3.2), we have

(3.5)

$$\begin{split}\frac{\mathrm{d}t}{\mathrm{d}\tau}&=\frac{b/\tau-1}{\partial f(t,w)/\partial t}=\frac{(\tau-b)t(1-t)[1-(1-t)w]}{\tau(1+b)w(t-t^{+}_{0}(w))(t-t^{-}_{0}(w))},\quad\quad\tau\neq b,\\ \frac{\mathrm{d}t}{\mathrm{d}\tau}&=\left\{\frac{t^{+}_{0}(w)(1-t^{+}_{0}(w))[1-(1-t^{+}_{0}(w))w]}{b\sqrt{1+4b^{2}(w-1)w}}\right\}^{1/2},\hskip 45.52458pt\tau=b,\end{split}$$

where we have used L’H$\hat{o}$pital’s rule for $\tau=b$ and

$$t_{0}^{\pm}(w)=\frac{2w-1\pm\sqrt{1+4b^{2}(w-1)w}}{2(1+b)w};$$

cf.(2.5). Coupling (2.1) and (3.2), we have, instead of (2.8),

(3.6)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n+1}}{2\pi i}\frac{\Gamma(n+N+2)e^{n}b^{-n}}{\Gamma(n+1)\Gamma(N-n+1)}\\ &\times\int_{0}^{+\infty}\int_{\gamma_{1}}\frac{1}{w-1}e^{Nf({t_{0}(w)},w)}e^{N(b\ln\tau-\tau)}\frac{\mathrm{d}t}{\mathrm{d}\tau}\mathrm{d}w\mathrm{d}\tau.\end{split}$$

Next, let us examine the mapping (2.12). We note that this mapping still works in the present case; the only difference is that for fixed $n/N\in(0,1)$, the constant $\eta$ in the mapping (2.12) is bounded away from 0, whereas when $n/N\rightarrow 0$, $\eta$ approaches 0. More precisely, we have $\eta\sim-b^{2}$. This can roughly be seen from the saddle points (2.9) and (2.11), together with their correspondence relation (2.13) under the mapping $w\rightarrow u$ given in (2.12). To prove this rigorously, we give the following lemma.

What Lemma 1 says is that if $a/b^{2}\rightarrow 0$ and $x=O(1)$ or $x\gg O(1)$, then we have $|\eta|N\sim b^{2}N=xb^{2}/a\rightarrow\infty$, i.e., $\eta N$ is large.

Coupling (3.6) and (2.12), we have

(3.12)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n+1}\Gamma(n+N+2)e^{n}b^{-n}e^{N\gamma}}{2\pi i\Gamma(n+1)\Gamma(N-n+1)}\\ &\times\int_{0}^{+\infty}\int_{\gamma_{1}}\frac{h(u,\tau)}{u-1}e^{N\psi(u)}e^{N(b\ln\tau-\tau)}\mathrm{d}w\mathrm{d}\tau,\end{split}$$

where

(3.13)

$$h(u,\tau)=\frac{u-1}{w-1}\frac{\mathrm{d}w}{\mathrm{d}u}\frac{\mathrm{d}t}{\mathrm{d}\tau},$$

$\mathrm{d}w/\mathrm{d}u$ is given by (2.14) and (2.15) and $\mathrm{d}t/\mathrm{d}\tau$ is given by (3.5). Following the same integration-by-parts procedure outlined in Section 2, we let $h_{0}(u,\tau)=h(u,\tau)$ in (3.13), and define $h_{l}(u,\tau)$ recursively by (2.18), (2.19) and (2.20). The final result is

	$\displaystyle t_{n}(x,N+1)=$	$\displaystyle\frac{(-1)^{n+1}\Gamma(n+N+2)e^{n}b^{-n}e^{N\gamma}}{\Gamma(N-n+1)N^{n+1}}$
(3.14)			$\displaystyle\times\Biggr{[}\mathbf{M}(aN+1,1,\eta N)\sum_{l=0}^{p-1}\frac{c_{l}}{(\eta N)^{l}}+\mathbf{M}^{\prime}(aN+1,1,\eta N)\sum_{l=0}^{p-1}\frac{d_{l}}{(\eta N)^{l}}+\varepsilon_{p}\Biggr{]},$

where

(3.15)		$\displaystyle c_{l}$	$\displaystyle=$	$\displaystyle\frac{N^{n+1}}{\Gamma(n+1)}\int_{0}^{\infty}a_{l}(\tau)\tau^{n}e^{-N\tau}\mathrm{d}\tau\sim\sum_{m=0}^{\infty}a_{l,m}\frac{\Gamma(n+m+1)}{\Gamma(n+1)N^{m}},$
(3.16)		$\displaystyle d_{l}$	$\displaystyle=$	$\displaystyle\frac{N^{n+1}}{\Gamma(n+1)}\int_{0}^{\infty}b_{l}(\tau)\tau^{n}e^{-N\tau}\mathrm{d}\tau\sim\sum_{m=0}^{\infty}b_{l,m}\frac{\Gamma(n+m+1)}{\Gamma(n+1)N^{m}}$

and

(3.17)

$$\varepsilon_{p}=\frac{N^{n+1}}{2\pi i\Gamma(n+1)(\eta N)^{p}}\int_{0}^{+\infty}\int_{\gamma_{1}}\frac{h_{p}(u,\tau)}{u-1}e^{N\psi(u)}e^{N(b\ln\tau-\tau)}\mathrm{d}w\mathrm{d}\tau.$$

Note that since $b=n/N\rightarrow 0$, the series in (3.15) and (3.16) are asymptotic. Moreover, since $\eta N$ is large in this case, (3) is indeed a compound asymptotic expansion as $N\rightarrow\infty$.

Now, we consider the subcase in which $a/b^{2}\rightarrow 0$ and $x=o(1)$. In this case, the expansion in (3) is still asymptotic as long as $|\eta|N\sim b^{2}N\rightarrow\infty$ or, equivalently, $n\gg O(\sqrt{N})$. If $n=O(\sqrt{N})$, then the series in (1.2) is itself an asymptotic expansion as $N\rightarrow\infty$. This completes our discussion of all three cases listed under the condition “$xN/n^{2}$ small” in Table 1.

For the case $a/b^{2}\in[\gamma,M]$, where $\gamma$ is a small positive number and $M$ is a large positive number, the saddle points $w_{\pm}$ in (2.9) are bounded away from the singularities $w=0$, $w=1$ and $w=\infty$, and coalesce with each other when $a/b^{2}\rightarrow\frac{1}{4(1-a)}$. Therefore, to derive an asymptotic expansion uniformly for $a/b^{2}\in[\gamma,M]$, we need the cubic transformation (see (4.6) below) introduced by Chester, Friedman and Ursell [2]. The resulting expansion is in terms of the Airy function $\text{Ai}(\cdot)$.

Following the same argument as in Section 3, we again use the mapping from $t\rightarrow\tau$ in (3.2), and start with the integral representation (3.6)

(4.1)

$$\begin{split}t_{n}(x,N+1)=&\frac{(-1)^{n+1}}{2\pi i}\frac{\Gamma(n+N+2)e^{n}b^{-n}}{\Gamma(n+1)\Gamma(N-n+1)}\\ &\times\int_{0}^{+\infty}\int_{\gamma_{1}}\frac{1}{w-1}e^{Nf({t_{0}(w)},w)}e^{N(b\ln\tau-\tau)}\frac{\mathrm{d}t}{\mathrm{d}\tau}\mathrm{d}w\mathrm{d}\tau,\end{split}$$

where the integration path is described in the line below (2.2). To proceed further, we divide $\gamma_{1}$ into two parts, and denote the part in the upper half of the plane by $\gamma^{(+)}$, and the other part in the lower half of the plane by $\gamma^{(-)}$. Recall from (2.2) that the phase function in (4.1) is given by

(4.2)

$$\begin{split}f(t_{0}(w),w)=&b\ln(1-t_{0}(w))+(1-b)\ln t_{0}(w)+a\ln w\\ &-a\ln(w-1)+b\ln[1-(1-t_{0}(w))w],\end{split}$$

which has a cut $(-\infty,1]$ in $w$-plane. Put

(4.3)

$$\begin{split}\overline{f}(t_{0}(w),w)=&b\ln(1-t_{0}(w))+(1-b)\ln t_{0}(w)+a\ln w\\ &-a\ln(1-w)+b\ln[1-(1-t_{0}(w))w],\end{split}$$

which has cuts $(-\infty,0]$ and $[1,\infty)$. From (4.2) and (4.3), we have

(4.4)

$$f(t_{0}(w),w)=\overline{f}(t_{0}(w),w)+a\pi i$$

for $w\in\gamma^{(+)}$, and

(4.5)

$$f(t_{0}(w),w)=\overline{f}(t_{0}(w),w)-a\pi i$$

for $w\in\gamma^{(-)}$. We now make the standard transformation

(4.6)

$$\overline{f}(t_{0}(w),w)=\frac{1}{3}u^{3}-\zeta u+A,$$

with the correspondence between the critical points of the two sides prescribed by

(4.7)

$$w_{+}\leftrightarrow\sqrt{\zeta},\qquad\quad w_{-}\leftrightarrow-\sqrt{\zeta}.$$

If $\zeta>0$, then $\pm\sqrt{\zeta}$ are both real; if $\zeta<0$, then $\pm\sqrt{\zeta}$ are complex conjugates and purely imaginary. The values of $A$ and $\zeta$ can be obtained by using (4.6) and (4.7). We also have

(4.8)

$$\begin{split}\frac{\mathrm{d}w}{\mathrm{d}u}&=\frac{(u-\sqrt{\zeta})(u+\sqrt{\zeta})w(1-w)\left[(2a-1)-\sqrt{1+4b^{2}w^{2}-4b^{2}w}\right]}{-2b^{2}(w-w_{+})(w-w_{-})},\quad u\neq\pm\sqrt{\zeta},\\ \frac{\mathrm{d}w}{\mathrm{d}u}&=\left\{\frac{2\sqrt{\zeta}a(1-2a)(1-a)}{b^{3}\sqrt{b^{2}-4a+4a^{2}}}\right\}^{1/2},\hskip 45.52458ptu=\pm\sqrt{\zeta},\end{split}$$

where we have again used L’H$\hat{o}$pital’s rule for $u=\pm\sqrt{\zeta}$. Let us first consider the case $\zeta<0$, and deform the image of the contour $\gamma^{(+)}$ under the mapping $w\rightarrow u$ defined in (4.6) to the steepest descent path of $\frac{1}{3}u^{3}-\zeta u+A$ in the $u$-plane which passes through $\sqrt{\zeta}$. We denote the path by $C_{2}$. Similarly, we deform the image of the contour $\gamma^{(-)}$, and denote the steepest descent path passing through $-\sqrt{\zeta}$ by $C_{3}$; see Figure 1. Next, we consider the case $\zeta>0$. Note that here the saddle points $\pm\sqrt{\zeta}$ are real, and that the contours $C_{2}$ and $C_{3}$ pass through both of them; see Figure 2. Clearly, in both cases, $C_{3}$ is the reflection of $C_{2}$ with respect to the real axis in the $u$-plane.