Fractional Powers of the Generating Function for the Partition Function

TL;DR

This paper explores congruences for coefficients of fractional powers of the generating function for the partition function, extending known results for integer powers to rational exponents.

Contribution

It introduces new congruences for $p_k(n)$ when $k$ is rational, generalizing classical partition congruences.

Findings

Identifies congruences for $p_k(n)$ with rational $k$

Extends classical partition congruences to fractional powers

Provides new insights into the structure of partition generating functions

Abstract

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discuss congruences satisfied by $p_{k}(n)$ when $k$ is a rational number.