Arithmetic Identities for Some Analogs of $5$-core Partition Function

TL;DR

This paper explores arithmetic identities related to analogs of the 5-core partition function, specifically focusing on functions a_5(n), b_5(n), and their relationships with the classical 5-core partition function.

Contribution

It introduces new arithmetic identities for the functions a_5(n) and b_5(n), expanding understanding of their properties and connections to 5-core partitions.

Findings

Derived several arithmetic identities for a_5(n) and b_5(n).

Analyzed relationships between a_5(n), b_5(n), and the classical 5-core partition function.

Enhanced the theoretical framework for analogs of core partition functions.

Abstract

Recently, Gireesh, Ray, and Shivashankar studied an analog, $\overline{a}_t(n)$, of the $t$-core partition function, $c_t(n)$. In this paper, we study the function $\overline{a}_5(n)$ in conjunction with $c_5(n)$ as well as another analogous function $\overline{b}_5(n)$. We also find several arithmetic identities for $\overline{a}_5(n)$ and $\overline{b}_5(n)$.