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

Abstract

Let ${\hat{I}}^2(q)={\sum }_{n\ge 0}\mathfrak{b}(n)q^n$ 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 $\mathfrak{b}(n)$.