Zeilberger's KOH theorem and the strict unimodality of q-binomial coefficients

TL;DR

This paper provides a new proof of the strict unimodality of q-binomial coefficients using Zeilberger's KOH theorem and demonstrates that the differences between consecutive coefficients can become arbitrarily large for fixed b and large a.

Contribution

It offers a Zeilberger's KOH theorem-based proof of strict unimodality and the unboundedness of coefficient differences, simplifying previous representation theory approaches.

Findings

Proof of strict unimodality using Zeilberger's KOH theorem.

Demonstration that coefficient differences can be arbitrarily large.

Simplification of previous proofs relying on representation theory.

Abstract

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\binom{a+b}{b}_q$ (see \cite{PP} and also \cite{PP2} for a slightly revised version of their theorem). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\binom{a+b}{b}_q$ can get arbitrarily large, when we assume that $b$ is fixed and $a$ is large enough.