Congruences for a class of eta-quotients and their applications

Abstract

The partition function $ p_{[1^c\ell^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}.\] In \cite{P}, we proved infinite family of congruences for this partition function for $\ell=11$. In this paper, we extend the ideas that we have used in \cite{P} to prove infinite families of congruences for the partition function $p_{[1^c\ell^d]}(n)$ modulo powers of $\ell$ for any integers $c$ and $d$, for primes $5\leq \ell\leq 17$. This generalizes Atkin, Gordon and Hughes' congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $\Gamma_0(\ell)$. Finally we used these congruences to prove congruences and incongruences of the generalized Frobenius $\ell$-color partitions, $\ell-$regular partitions and $\ell-$core partitions for $\ell=5,7,11,13$ and $17$.