HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

Abstract

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).