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 $c(\ell n+a)\equiv 0(\mathrm{mod}\ell )$. 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.

12 pages. V2: Minor typographical and mathematical corrections

Cited by in corpus (1)

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