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arXiv · 1203.6736

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

Guoniu Han·Frédéric Jouhet·Jiang Zeng
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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.

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Guoniu Han, Frédéric Jouhet, Jiang Zeng. 2012-03-30. Two new triangles of $q$-integers via $q$-Eulerian polynomials of type $A$ and $B$. https://arxiv.org/abs/1203.6736

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