Congruence properties modulo prime powers for a class of partition functions

Abstract

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n{)}^{-1}(1-q^{pn}{)}^{-1}$. This function and its generalizations $p_{[c^{\ell },d^m]}(n)$ are the subject of study in several recent papers. Let $\ell \ge 5$, let $j\ge 1$, and let $p\in \{2,3,5\}$. In this paper, we prove that the generating function for $p_{[1,p]}(n)$ in the progression ${\hat{I}}_{p,\ell ,j}^2$ modulo ${\ell }^j$ with $24{\hat{I}}_{p,\ell ,j}^2\equiv p+1(\mathrm{mod}{\ell }^j)$ lies in a Hecke-invariant subspace of type $\{\hat{I}\cdot (Dz)\hat{I}\cdot (Dpz)F(Dz):F(z)\in M_s(\hat{I}{“}_0(p),\ddot{I}‡)\}$ for suitable $D\ge 1$, $s\ge 0$, and character~$\ddot{I}‡$. When $p\in \{2,3,5\}$, we use the Hecke-invariance of these subspaces proved in [21] to prove, for distinct primes $\ell$ and $m\ge 5$ and $j\ge 1$, congruences of the form \[ p_{[1, p]}\left(\frac{\ell^jm^k n + 1}{D}\right)\equiv 0 \pmod{\ell^j} \] for all $n\ge 1$ with $m\nmid n$, where $k$ is explicitly computable and depends on the forms in the invariant subspace. Our proofs require adapting and extending analogous level one results on $p(n)$ in [1] and [22] to level $p$.

Simplified Thm. 1.3, added Thm. 3.3 and Prop. 3.4, and revised proof of Thm. 1.3. To appear in Research in Number Theory