Linear identities for partition pairs with $4$-cores

TL;DR

The paper investigates A_4(n), the count of partition pairs with 4-cores, via elementary $q$-series and 3-dissection techniques to produce infinite families of linear identities and congruences. It establishes explicit generating functions for $A_4(3n+1)$ and $A_4(27n+19)$, then proves that for all $k\ge0$ and $n$ with $3\nmid n$, $$A_4\left(3^{2k+1}n+\frac{3^{2k+2}-5}{4}\right)=\frac{(3^{2k}-1)(3^{2k+2}-1)}{640}A_4(27n+19) -\frac{(3^{2k}-9)(3^{2k+2}-1)}{64}A_4(3n+1).$$ This yields infinite congruences and demonstrates a structural relation among indices $3n+1$, $27n+19$, and the powers of 3, all derived without heavy modular form machinery. The results also show congruence consequences such as $A_4(27n+19) \equiv 0 \pmod{492}$ when $3 \nmid n$, and provide a framework to analyze further linear identities via $U$ and $V$ operators on $q$-series. These contributions advance understanding of 4-core partition pairs and their arithmetic properties through explicit theta-function identities and dissections.

Abstract

We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities.

Theorems & Definitions (31)

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