Ramanujan congruences for fractional partition functions

Abstract:

For rational $\alpha$, the fractional partition functions $p_\alpha (n)$ are given by the coefficients of the generating function $(q; q)_\infty ^\alpha$. When $\alpha = -1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod {\ell }$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell \in \{5, 7, 11\}$. Chan and Wang recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as $p_\frac {57}{61}(17^2 n - 3) \equiv 0 \pmod {17^2}$.