An infinite family of congruences arising from a second order mock theta function

TL;DR

This paper discovers an infinite family of congruences modulo powers of 3 for the coefficients of a specific second order mock theta function, expanding understanding of their arithmetic properties.

Contribution

It introduces a new infinite family of congruences modulo powers of 3 for the coefficients of a second order mock theta function.

Findings

Established congruences modulo powers of 3 for $eta(n)$ coefficients.

Extended the arithmetic understanding of second order mock theta functions.

Provided methods for deriving infinite congruence families.

Abstract

Let $\beta(q)=\sum_{n\ge 0} \mathfrak{b}(n)q^n$ be a second order mock theta function defined by $$\sum_{n\ge 0}\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)_{n+1}^2}.$$ In this paper, we obtain an infinite family of congruences modulo powers of $3$ for $\mathfrak{b}(n)$.