Hecke-type congruences for two smallest parts functions

TL;DR

This paper establishes infinite families of congruences modulo small primes for overpartition and smallest parts functions, extending classical Hecke-type congruence results to new partition-related functions.

Contribution

It introduces new Hecke-type congruences for overpartition and smallest parts functions, using eigenforms and Hecke operators, expanding the scope of classical partition congruence results.

Findings

Proves infinite congruences modulo 3, 5, and powers of 2.

Establishes relations between generating functions and eigenforms.

Extends classical Hecke-type congruences to new partition functions.

Abstract

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\bar{p}(n)$ and two smallest parts functions: $\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$ 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 $\operatorname{spt}(n)$ in 2010. The proofs depend on congruences between the generating functions for $\bar{p}(n)$, $\bar{\operatorname{spt1}}(n),$ and $\operatorname{M2spt}(n)$ and eigenforms for the half-integral weight Hecke operator $T(\ell^{2})$.