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

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 $F_{a,c}(ζ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- ζ{\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $ζ$ 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.

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