Linear identities for partition pairs with $5$-cores

TL;DR

This paper establishes an infinite family of linear identities for counting partition pairs with 5-cores, leading to new congruences, by leveraging theta function identities involving Ramanujan's parameter.

Contribution

It introduces novel linear identities for $A_5(n)$ using theta functions, enabling derivation of new congruences for partition pairs with 5-cores.

Findings

Derived infinite linear identities for $A_5(n)$

Established new congruences for partition pairs with 5-cores

Utilized theta function identities involving Ramanujan's parameter

Abstract

We prove an infinite family of linear identities for the number $A_5(n)$ of partition pairs of $n$ with $5$-cores by using certain theta function identities involving the Ramanujan's parameter $k(q)$ due to Cooper, and Lee and Park. Consequently, we deduce an infinite family of congruences for $A_5(n)$ using these linear identities.