A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5

TL;DR

This paper presents a novel single-variable algebraic proof of an infinite family of congruences related to the smallest parts function, simplifying previous multi-relation induction methods and applying modular function theory in a new way.

Contribution

It introduces a single-variable approach using a localized ring of modular functions to prove a family of partition congruences, extending algebraic methods in the field.

Findings

Proof requires only 10 initial relations

Utilizes a ring of modular functions isomorphic to a localization of Z[X]

First application of such algebraic structure to partition congruences

Abstract

In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.