Strong q-log-convexity of the Eulerian polynomials of Coxeter groups

TL;DR

This paper proves the strong q-log-convexity of Eulerian polynomials for Coxeter groups using exponential generating functions and continued fractions, providing a unified approach for various types.

Contribution

It introduces a new proof technique for strong q-log-convexity of Eulerian polynomials via exponential Riordan arrays and continued fractions.

Findings

Proves strong q-log-convexity for Eulerian polynomials of types A and B

Extends results to q-analogues and generalized Eulerian polynomials

Provides a unified framework for different Eulerian polynomial families

Abstract

In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unified manner.