A note on restricted partition functions of Pushpa and Vasuki

TL;DR

This work advances the 2-adic congruence theory for three restricted partition functions $M(n), T^\ast(n), P^\ast(n)$ by establishing infinite families of congruences modulo powers of $2$ using elementary $q$-series techniques. The authors develop theta-function identities and shift-index generating functions, notably expressing $\sum_{n=-1}^\infty M(2n+3)q^n$, $\sum_{n=-1}^\infty T^\ast(2n+2)q^n$, and $\sum_{n=-1}^\infty P^\ast(2n+3)q^n$ in terms of Euler products $f_m$ and related series. They then derive generating functions for $M(2^k n+2^{k+2}-1)$, $T^\ast(2^k n+2^{k+2}-2)$, and $P^\ast(2^k n+2^{k+2}-1)$ involving recurrences for auxiliary sequences $A_k,B_k,C_k$, with $P_k=-4P_{k-1}-8P_{k-2}+5\cdot 2^{k-1}$ and $C_k=-4C_{k-1}-8C_{k-2}$. The main results are two theorems: (i) certain $2$-adic divisibilities for $M(n)$ and $T^\ast(n)$ and, (ii) detailed $2$-adic congruences for $P^\ast(n)$ across multiple residue classes, including explicit expressions in terms of $(-64)^k$ for scaled subsequences. These findings extend known congruence phenomena for restricted partitions and showcase how elementary $q$-series can yield deep arithmetic structure.

Abstract

We establish infinite families of congruences modulo arbitrary powers of $2$ for three restricted partition functions $M(n), T^\ast(n)$, and $P^\ast(n)$ recently introduced by Pushpa and Vasuki by employing elementary $q$-series techniques.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof