The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

TL;DR

This paper develops a continued fraction approach to derive symmetric and unimodal expansions of Eulerian polynomials, proving conjectures and extending results to derangements and $(p,q)$-analogues, unifying previous findings.

Contribution

It introduces a continued fraction framework to prove symmetry and unimodality of Eulerian polynomial expansions and extends these results to derangements and $(p,q)$-analogues.

Findings

Derived a continued fraction formula confirming the conjectured properties.

Established a $(p,q)$-analogue unifying previous results.

Proved that (-1)-evaluations relate to tangent and secant numbers.

Abstract

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a $(p,q)$-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The $(p,q)$-analogue unifies and generalizes our recent results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).