Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$

Abstract

The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n, \end{equation*} have been extensively studied. For instance, for all $n\ge0$, Kaur and Rana established congruences such as for all $n\ge0$, \begin{align*} b(12n+10) &\equiv 0 \pmod{36}, \quad b(18n+16) \equiv 0 \pmod{72}, \end{align*} Chen and Mao proved that for all $n\ge0$, \begin{align*} b(4n+1) &\equiv 0 \pmod{2}, \quad b(4n+2) \equiv 0 \pmod{4}, \end{align*} while Mao also showed that for all $n\ge0$, \begin{align*} b(6n+2) &\equiv 0 \pmod{4}, \quad b(6n+4) \equiv 0 \pmod{9}. \end{align*} In this paper, we find new congruences and infinite families of congruences modulo $2, 4, 8, 36, 54, 72$ for the function $\mathcal{B}(q)$. For example, let $p \geq 5$ be a prime, if $\left(\frac{-3}{p}\right)_L = -1$, then for all $n, k \geq 0$ with $p \nmid n$, we have \begin{equation*} b\left( 3p^{2k+1}n + \frac{p^{2k+2}-1}{2} \right) \equiv 0 \pmod{2}. \end{equation*} Let $p \geq 5$ be a prime and $1 \leq \ell \leq p - 1$ such that $\left( \frac{12\ell + 9}{p} \right)_L = -1$. Then for all $n, k \geq 0$, we have \begin{equation*} b\left(6p^{2k+3}n + \frac{3p^{2k+2}(4\ell+3)-1}{2}\right) \equiv 0 \pmod{36}. \end{equation*} Our techniques involve elementary $q$-series and Maple.