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 Fa,c(ζ;e−z):=∏n≥0(1−ζe−(a+cn)z)𝐹𝑎,𝑐(𝜁;e−𝑧):=∏𝑛≥0(1−𝜁e−(𝑎+𝑐𝑛)𝑧) (a,c∈N𝑎,𝑐∈𝑁 with 00 < a≤c𝑎≤𝑐 and ζ𝜁 a root of unity) when z𝑧 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.