Some identities on the second order mock theta functions

TL;DR

This paper provides analytic proofs for three identities involving second order mock theta functions, addressing an open problem posed by Nath and Das, using theta function parametrization and known identities.

Contribution

It offers the first analytic proofs of specific identities on second order mock theta functions, advancing understanding in mock theta function theory.

Findings

Proved three identities on second order mock theta functions.

Utilized $(p, k)$-parametrization of theta functions.

Applied identities from Hickerson and Mortenson.

Abstract

Recently, Nath and Das investigated congruence properties for the second order mock theta function $B(q)$. In their paper, they asked for analytic proofs of three identities on the second order mock theta functions $A(q)$, $B(q)$ and $\mu_2(q)$. In this paper, we settle Nath and Das' open problem by using the $(p, k)$-parametrization of theta functions and several identities due to Hickerson and Mortenson.