A Note on the Equidistribution of 3-Colour Partitions - Males, Joshua (Symmetry, Integrability and Geometry: Methods and Applications, 2024)
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 ₐ,c(ζ; e⁻ᶻ) := ∏ₙ≥₀ (1 − ζe⁻⁽ᵃ⁺ᶜⁿ⁾ᶻ) (, c ∈ ℕ with 0 < ≤ c and ζ a root of unity) when 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.