Fractional Powers of the Generating Function for the Partition Function

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 $p(5n+4)\equiv 0(\mathrm{mod}5)$. In this article, we discuss congruences satisfied by $p_k(n)$ when $k$ is a rational number.