Infinite Families of Congruences Modulo 5 for Ramanujan's General Partition Function

TL;DR

This paper establishes new infinite families of congruences modulo 5 for Ramanujan's general partition function using q-identities, extending known results to various residue classes and parameters.

Contribution

It introduces novel Ramanujan-type congruences modulo 5 for the general partition function for multiple residue classes and parameters, expanding the understanding of its modular properties.

Findings

New congruences modulo 5 for p_r(n) for specific r-values.

Infinite families of congruences parametrized by λ.

Application of q-identities to derive these congruences.

Abstract

For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5\lambda+1), -(5\lambda+3), -(5\lambda+4), -(25\lambda+1)$, $-(25\lambda+2)$, and any integer $\lambda$.