Polynomials with palindromic and unimodal coefficients

TL;DR

This paper studies palindromic polynomials with unimodal coefficients, providing basis transition matrices, characterizations, and conditions for nonnegative coefficients, and explores their connection to rank-generating functions of posets.

Contribution

It introduces new basis transition matrices for palindromic polynomials and characterizes those with nonnegative coefficients, linking them to poset rank-generating functions.

Findings

Derived transition matrices between bases of palindromic polynomials.

Provided characterizations for polynomials with nonnegative coefficients.

Linked palindromic polynomials to rank-generating functions of posets.

Abstract

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace $\mathcal{P}_n(q)$ of $\mathbb{R}(q)_{n+1}$ of dimension $\lfloor{n/2}\rfloor+1$. We give transition matrices between two bases $\left\{q^j(1+q+\cdots+q^{n-2j})\right\}, \left\{q^j(1+q)^{n-2j}\right\}$ and the standard basis $\left\{q^j(1+q^{n-2j})\right\}$ of $\mathcal{P}_n(q)$. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.