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.
Categories: math.CO math.NT