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On a polynomial congruence for Eulerian polynomials

Ira M. Gessel∗ Department of Mathematics
Brandeis University
Waltham, MA 02453 gessel@brandeis.edu
(Date: January 18, 2021)
Supported by a grant from the Simons Foundation (#427060, Ira Gessel).

Yoshinaga [2, Proposition 5.5] proved, using arrangements of hyperplanes, the polynomial congruence for Eulerian polynomials

An​(tm)≡(1+t+⋯+tm−1m)n+1​An​(t)mod(t−1)n+1.A_{n}(t^{m})\equiv\left(\frac{1+t+\cdots+t^{m-1}}{m}\right)^{n+1}A_{n}(t)\mod(t-1)^{n+1}. (1)

Here the Eulerian polynomials An​(t)A_{n}(t) may be defined by the generating function

∑n=0∞An​(t)(1−t)n+1​xnn!=11−t​ex.\sum_{n=0}^{\infty}\frac{A_{n}(t)}{(1-t)^{n+1}}\frac{x^{n}}{n!}=\frac{1}{1-te^{x}}. (2)

A simpler proof, using roots of unity, was given by Iijima et al. [1]. We give here a very simple proof based on the generating function (2).

Since 1+t+⋯+tm−1=(1−tm)/(1−t)1+t+\cdots+t^{m-1}=(1-t^{m})/(1-t), the congruence (1) is equivalent to the statement that the denominator of

mn+1​An​(tm)(1−tm)n+1−An​(t)(1−t)n+1m^{n+1}\frac{A_{n}(t^{m})}{(1-t^{m})^{n+1}}-\frac{A_{n}(t)}{(1-t)^{n+1}} (3)

is not divisible by t−1t-1.

By (2) the rational function (3) is the coefficient of xn/n!x^{n}/n! in

m1−tm​em​x−11−t​ex=m1−tm​em​x−1+t​ex+t2​e2​x+⋯+tm−1​e(m−1)​x1−tm​em​x.\frac{m}{1-t^{m}e^{mx}}-\frac{1}{1-te^{x}}=\frac{m}{1-t^{m}e^{mx}}-\frac{1+te^{x}+t^{2}e^{2x}+\cdots+t^{m-1}e^{(m-1)x}}{1-t^{m}e^{mx}}.

Thus it suffices to show that the denominator of the coefficient of xn/n!x^{n}/n! in

1−tj​ej​x1−tm​em​x=1+t​ex+⋯+tj−1​e(j−1)​x1+t​ex+⋯+tm−1​e(m−1)​x\frac{1-t^{j}e^{jx}}{1-t^{m}e^{mx}}=\frac{1+te^{x}+\cdots+t^{j-1}e^{(j-1)x}}{1+te^{x}+\cdots+t^{m-1}e^{(m-1)x}} (4)

is not divisible by t−1t-1. We have 1+t​ex+⋯+tm−1​e(m−1)​x=1+t+⋯+tm−1+x​P​(t,x)1+te^{x}+\cdots+t^{m-1}e^{(m-1)x}=1+t+\cdots+t^{m-1}+xP(t,x) where P​(t,x)P(t,x) is a power series in xx with coefficients that are polynomials in tt. It follows that the denominator of the coefficient of xn/n!x^{n}/n! in in (4) is a constant times a power of 1+t+⋯+tm−11+t+\cdots+t^{m-1} and is thus not divisible by t−1t-1.

References

  • [1] Kazuki Iijima, Kyouhei Sasaki, Yuuki Takahashi, and Masahiko Yoshinaga, Eulerian polynomials and polynomial congruences, Contrib. Discrete Math. 14 (2019), no. 1, 46–54.
  • [2] Masahiko Yoshinaga, Worpitzky partitions for root systems and characteristic quasi-polynomials, Tohoku Math. J. (2) 70 (2018), no. 1, 39–63.