Congruences for the coefficients of the Gordon and McIntosh mock theta function $\xi(q)$

TL;DR

This paper investigates the arithmetic properties of the coefficients of Gordon and McIntosh's third order mock theta function (q), revealing several Ramanujan-like congruences and advancing understanding of its number-theoretic structure.

Contribution

The paper introduces new arithmetic properties and infinite families of congruences for the coefficients of (q), expanding the knowledge of mock theta functions' arithmetic behavior.

Findings

Multiple Ramanujan-like congruences for coefficients

Identification of infinite families of congruences

Enhanced understanding of mock theta function arithmetic

Abstract

Recently Gordon and McIntosh introduced the third order mock theta function $\xi(q)$ defined by $$ \xi(q)=1+2\sum_{n=1}^{\infty}\frac{q^{6n^2-6n+1}}{(q;q^6)_{n}(q^5;q^6)_{n}}. $$ Our goal in this paper is to study arithmetic properties of the coefficients of this function. We present a number of such properties, including several infinite families of Ramanujan--like congruences.