On second order mock theta function $ B(q) $

<abstract><p>In this paper, we present some arithmetic properties for the second order mock theta function $ B(q) $ given by McIntosh as:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ B(q) = \sum\limits_{n = 0}^{\infty}\frac{q^n(-q;q^2)_n}{(q;q^2)_{n+1}}. $\end{document} </tex-math></disp-formula></p> </abstract>

Combinatorics of two second-order mock theta functions

We introduce combinatorial interpretations of the coefficients of two second-order mock theta functions. Then, we provide a bijection that relates the two combinatorial interpretations for each function. By studying other special cases of the multivariate identity proved by the bijection, we obtain new combinatorial interpretations for the coefficients of Watson’s third-order mock theta function [Formula: see text] and Ramanujan’s third-order mock theta function [Formula: see text].

OVERPARTITIONS RELATED TO THE MOCK THETA FUNCTION

Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number $v_{0}(n)$ of overpartitions of $n$ into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function $V_{0}(q)$ of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of $v_{0}(2n)$. Then we establish three congruences for $v_{0}(n)$ along certain progressions which are subsequences of the integers $4n+3$.

ARITHMETIC PROPERTIES OF COEFFICIENTS OF THE MOCK THETA FUNCTION

We investigate the arithmetic properties of the second-order mock theta function $B(q)$ and establish two identities for the coefficients of this function along arithmetic progressions. As applications, we prove several congruences for these coefficients.

A Mock Theta Function of Second Order

We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .