Congruences for the Coefficients of the Powers of the Euler Product

TL;DR

This paper derives new congruences for the coefficients of powers of the Euler product, revealing deep modular properties and enabling the discovery of infinite congruence families for various partition functions.

Contribution

It introduces novel methods using modular equations and linear recurring sequences to establish broad families of congruences for partition-related functions.

Findings

Derived generating functions for specific $p_{8k}$ and $p_{3k}$ coefficients.

Established infinite families of congruences for $p_k(n)$ modulo any $m extgreater 1$.

Extended congruence results to overpartition, $t$-core, and $ ext{l}$-regular partition functions.

Abstract

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\alpha} n +\frac{k(2^{2\alpha}-1)}{3})$ $(1\leq k\leq 3)$ and $p_{3k} (3^{2\beta}n+\frac{k(3^{2\beta}-1)}{8})$ $(1\leq k\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite families of congruences for $p_k(n)$ modulo any $m\geq2$, where $1\leq k\leq 24$ and $3|k$ or $8|k$. Based on these congruences for $p_k(n)$, infinite families of congruences for many partition functions such as the overpartition function, $t$-core partition functions and $\ell$-regular partition functions are easily obtained.