Ramanujan congruences for a class of eta quotients

Abstract

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes $\ell$ for which their coefficients $c(n)$ obey congruences of the form $c(\ell n + a) \equiv 0 \pmod \ell$. We apply this result to obtain a complete characterization of the congruences of the same form that the sequences $c_N(n)$ satisfy, where $c_N(n)$ is defined by $ \sum_{n=0}^{\infty} c_N(n)q^n = \prod_{n=1}^{\infty} \frac{1}{(1-q^n)(1-q^{Nn})}$. This last result answers a question of H.-C. Chan.