Lacunary Eta-quotients Modulo Powers of Primes

TL;DR

This paper investigates the lacunarity of eta-quotients modulo prime powers, extending previous work on partition generating functions and their divisibility properties.

Contribution

It generalizes prior results on lacunarity conditions for eta-quotients and applies these to various partition-related generating functions.

Findings

Established new criteria for lacunarity of eta-quotients modulo prime powers

Extended previous results to generalized Dedekind eta-quotients

Applied findings to partition functions studied by Nekrasov, Okounkov, and Han

Abstract

An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Okounkov, and Han.