Congruences between the coefficients of certain mock theta functions

Abstract

In this article, we prove several congruences modulo 3, 4, 5, 8, 9, 12, 24, 27, 81, 243, and 729 enjoyed by the coefficients of certain mock theta functions. As an example, for the second order mock theta functions${\mu }_2\left(q\right):=\sum _{n=0}^{\infty }\frac{{(-1)}^n{q}^{n^2}{(q;q^2)}_n}{{(-q^2;q^2)}_n^2}=\sum _{n=0}^{\infty }{P}_{{\mu }_2}\left(n\right)q^n,B_2\left(q\right):=\sum _{n=0}^{\infty }\frac{q^{n(n+1)}{(-q^2;q^2)}_n}{{(q;q^2)}_{n+1}^2}=\sum _{n=0}^{\infty }\frac{q^n{(-q;q^2)}_n}{{(q;q^2)}_{n+1}}=\sum _{n=0}^{\infty }{P}_{B_2}\left(n\right)q^n,$ we have$P_{{\mu }_2}(27n+26)\equiv 25P_{B_2}(108n+103)\left(\mathrm{mod}27\right)$ for all $n\ge 0$.