The Limiting Distribution of the Coefficients of the $q$-Catalan Numbers

TL;DR

This paper proves that the coefficients of the $q$-Catalan numbers tend to follow a normal distribution as n grows large, and discusses their unimodality and log-concavity properties.

Contribution

It establishes the asymptotic normality of the coefficients of $q$-Catalan numbers and conjectures their unimodality and log-concavity for large n.

Findings

Coefficients follow a normal distribution asymptotically.

Unimodality and log-concavity are conjectured for large n.

Coefficients are not unimodal for small n.

Abstract

We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.