Some properties of a class of refined Eulerian polynomials

TL;DR

This paper introduces a new exponential generating function for refined Eulerian polynomials, providing explicit formulas and connections to Eulerian and Catalan number generating functions, revealing new relations between Eulerian and Euler numbers.

Contribution

It derives an explicit formula for refined Eulerian polynomials in terms of classical Eulerian and Catalan generating functions and establishes new relations between Eulerian and Euler numbers.

Findings

Derived exponential generating function for A_n(p,q)

Expressed A_n(p,q) explicitly using Eulerian and Catalan generating functions

Connected A_n(p,q) with classical Eulerian and Euler numbers

Abstract

In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely, \begin{eqnarray*} A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)} \end{eqnarray*} for $n\geq 1$, where ${odes}(\pi)$ and ${edes}(\pi)$ enumerate the number of descents of permutation $\pi$ in odd and even positions, respectively. In this paper, we build an exponential generating function for $A_{n}(p,q)$ and establish an explicit formula for $A_{n}(p,q)$ in terms of Eulerian polynomials $A_{n}(q)$ and $C(q)$, the generating function for Catalan numbers. In certain special case, we set up a connection between $A_{n}(p,q)$ and $A_{n}(p,0)$ or $A_{n}(0,q)$, and express the coefficients of $A_{n}(0,q)$ by Eulerian numbers. Specially, this connection creates a new relation between Euler numbers and Eulerian numbers.