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 $F_{a,c}(\zeta ;{\mathrm{e}}^{-z}):={\prod }_{n\ge 0}(1-\zeta {\mathrm{e}}^{-(a+cn)z})$ ($a,c\in \mathbb{N}$ with $0<a\le c$ 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.