The ratio monotonicity of Eulerian-type polynomials

Abstract

This paper is motivated by determining the location of modes of some unimodal Eulerian-type polynomials. The notion of ratio monotonicity was introduced by Chen-Xia when they investigated the $q$-derangement numbers. Let $(f_n(x){)}_{n⩾0}$ be a sequence of real polynomials satisfying the Eulerian-type recurrence relation $$f_{n+1}(x)=(anx+bx+c)f_n(x)+ax(1-x)\frac{\mathrm{d}}{\mathrm{d}x}{f}_n(x),f_0(x)=1,$$ where $a,b$ and $c$ are nonnegative integers. Assume that $deg{f}_n(x)=n$. Setting $g_n(x)=x^n{f}_n\left(\frac{1}{x}\right)$, we have $$g_{n+1}(x)=(anx+b+cx)g_n(x)+ax(1-x)\frac{\mathrm{d}}{\mathrm{d}x}{g}_n(x),$$ We find that if $a+c⩾b⩾c>0$, then $f_n(x)$ is bi-gamma-positive and $g_n(x)$ is ratio monotone. As applications, we discover the ratio monotonicity of several Eulerian-type polynomials, including the $(exc,cyc)$ $q$-Eulerian polynomials, the $1/k$-Eulerian polynomials, a kind of generalized Eulerian polynomials studied by Carlitz-Scoville, the $(de{s}_B,neg)$ $q$-Eulerian polynomials over the hyperoctahedral group and the $r$-colored Eulerian polynomials. In particular, let $A_n(x,q)$ be the $(exc,cyc)$ $q$-Eulerian polynomials, we find that the polynomials $x^{n-1}{A}_n(1/x,q)$ are ratio monotone when $0<q⩽1$, while $A_n(x,q)$ are ratio monotone when $1⩽q⩽2$.

14 pages