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

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.