A note on the equidistribution of $3$-colour partitions

TL;DR

This paper establishes equidistribution results for three-color partitions by deriving asymptotic formulas for related infinite products, advancing understanding of their asymptotic behavior and distribution properties.

Contribution

It introduces new asymptotic formulas for infinite products associated with three-color partitions, enabling analysis of their equidistribution and asymptotic behavior.

Findings

Proved asymptotic formulas for specific infinite products

Established equidistribution results for three-color partition families

Derived asymptotic behavior of these partitions

Abstract

In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(\zeta ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- \zeta {\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $\zeta$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.