Congruences modulo powers of 11 for some eta-quotients

Abstract

The partition function $$ p_{[1^c11^d]}(n)$$ can be defined using the generating function, $$\begin{aligned} \sum _{n=0}^{\infty }p_{[1^c{11}^d]}(n)q^n=\prod _{n=1}^{\infty }\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}. \end{aligned}$$In this paper, we prove infinite families of congruences for the partition function $$ p_{[1^c11^d]}(n)$$ modulo powers of 11 for any integers c and d, which generalizes Atkin and Gordon’s congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $$\Gamma _0(11)$$.