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.
Categories: math.NT