Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 001, 8 pages      arXiv:2307.12955      https://doi.org/10.3842/SIGMA.2024.001
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

A Note on the Equidistribution of 3-Colour Partitions

Joshua Males ^ab
a) School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
^b) Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, BS8 1UG, UK

Received July 25, 2023, in final form December 28, 2023; Published online January 01, 2024

Abstract
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.

Key words: asymptotics; partitions; Wright's circle method.

pdf (382 kb)   tex (12 kb)  

References

1. Bringmann K., Craig W., Males J., Ono K., Distributions on partitions arising from Hilbert schemes and hook lengths, Forum Math. Sigma 10 (2022), e49, 30 pages, arXiv:2109.10394.
2. Campbell R., Les intégrales eulériennes et leurs applications. Étude approfondie de la fonction gamma, Collection Universitaire de Mathématiques, Vol. 20, Dunod, Paris, 1966.
3. Chern S., Nonmodular infinite products and a conjecture of Seo and Yee, Adv. Math. 417 (2023), 108932, 40 pages, arXiv:1912.10341.
4. Ciolan A., Equidistribution and inequalities for partitions into powers, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 66 (2023), 409-431, arXiv:2002.05682.
5. Craig W., Seaweed algebras and the index statistic for partitions, J. Math. Anal. Appl. 528 (2023), 127544, 23 pages, arXiv:2112.09269.
6. Hardy G.H., Ramanujan S., Asymptotic formulae in combinatory analysis, Proc. Lond. Math. Soc. 17 (1918), 75-115.
7. Liu Z.-G., Zhou N.H., Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions, Ramanujan J. 57 (2022), 1085-1123, arXiv:1903.00835.
8. Ngo H.T., Rhoades R.C., Integer partitions, probabilities and quantum modular forms, Res. Math. Sci. 4 (2017), 17, 36 pages.
9. Schlosser M.J., Zhou N.H., Expansions of averaged truncations of basic hypergeometric series, arXiv:2307.10821.
10. Wright E.M., Stacks, Quart. J. Math. Oxford Ser. (2) 19 (1968), 313-320.
11. Wright E.M., Stacks. II, Quart. J. Math. Oxford Ser. (2) 22 (1971), 107-116.
12. Zhou N.H., Note on partitions into polynomials with number of parts in an arithmetic progression, Int. J. Number Theory 17 (2021), 1951-1963, arXiv:1909.13549.