q-Binomials and related symmetric unimodal polynomials

TL;DR

This paper explores the history and proofs of the unimodality of q-binomial coefficients, introduces perturbations to recurrences to generate new unimodal polynomials, and analyzes their properties.

Contribution

It presents new perturbations to existing recurrences, expanding the family of unimodal polynomials and analyzing their behavior.

Findings

Perturbations can produce a larger family of unimodal polynomials.

Analysis of specific cases reveals how perturbations affect unimodality.

The recurrence approach offers an elegant framework for generating unimodal polynomials.

Abstract

The q-binomial coefficients were assumed to be unimodal as early as the 1850's, but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an "elementary" proof using linear algebra. Finally, in 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyze how these perturbations affect the final polynomial and analyze some specific cases.