The MacMahon $q$-Catalan is convex

TL;DR

This paper investigates the convexity properties of MacMahon's $q$-Catalan polynomials, providing new inequalities and insights into their behavior as functions of $q$, with implications for partition generating functions.

Contribution

It establishes the convexity of MacMahon's $q$-Catalan polynomials and presents new inequalities related to their properties and partition generating functions.

Findings

Proves convexity of MacMahon's $q$-Catalan polynomials.

Introduces new inequalities for partition generating functions.

Provides insights into the mathematical structure of $q$-Catalan polynomials.

Abstract

Let $n\geq2$ be an integer. In this paper, we study the convexity of the so-called MacMahon's $q$-Catalan polynomials $C_n(q)=\frac1{[n+1]_q}\left[ 2n \atop n \right]_q$ as functions of $q$. Along the way, several intermediate results on inequalities are presented including a commentary on the convexity of the generation function for the integer partitions.