Linear identities for partition pairs with $5$-cores

TL;DR

This work analyzes $A_5(n)$, the count of partition pairs with both parts as $5$-cores, by developing elementary $q$-series methods centered on Ramanujan's theta-parameter $k(q)$. It derives an exact generating function for the subsequence $A_5(2^k n+2^{k+1}-2)$ in terms of theta-products and a recurrence sequence $B_k$ defined by $B_0=0$, $B_1=1$, and $B_k=-4B_{k-1}-8B_{k-2}+(8^k-1)/7$, enabling a family of linear identities for $A_5(n)$ and, consequently, infinite congruences. The approach relies on carefully orchestrated $2$-dissection formulas and $k(q)$-related identities to manipulate generating functions and extract coefficients. A key outcome is the linear relation $A_5(2^{k+1}n+3\cdot 2^k-2)=B_k A_5(4n+4)+\left((8^{k+1}-1)/7-9B_k\right)A_5(2n+1)$, which yields explicit congruences such as $A_5(2^{4m+4}n+3\cdot 2^{4m+3}-2)\equiv 0 \pmod{(8^{4m+4}-1)/91}$. The results advance the arithmetic of $5$-core partition pairs through a structured, theta-function-driven framework.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5