Two new triangles of $q$-integers via $q$-Eulerian polynomials of type $A$ and $B$

TL;DR

This paper introduces new $q$-analogues of Eulerian polynomial expansions for types A and B, revealing new polynomial sequences with positive coefficients and exploring their properties and potential combinatorial interpretations.

Contribution

It provides the first $q$-analogues of certain Eulerian polynomial expansions for types A and B, along with new polynomial sequences with positive coefficients.

Findings

Established $q$-analogues of Eulerian polynomial expansions

Derived new polynomial sequences with positive integral coefficients

Proposed open problem for combinatorial interpretation

Abstract

The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open problem is to give a combinatorial interpretation for these polynomials.