Some identities on the second order mock theta functions
Xingyuan Cai, Eric H. Liu, Olivia X. M. Yao
TL;DR
This work provides complete analytic proofs for three identities relating the second order mock theta functions $A(q)$, $B(q)$, and $\mu_2(q)$, addressing Nath and Das' open problem. The authors employ the $$(p,k)$$-parametrization of theta functions and Hickerson–Mortenson Appell–Lerch identities to translate the $q$-series into structured theta quotients and $m$-sum expressions, enabling modular-coefficient extraction on the $3n$ and $3n+1$ subsequences. Central to the approach are auxiliary zero-relations $F_1(q)=0$, $F_2(q)=0$, and $F_3(q)=0$, established via intricate manipulations of product identities and substitutions; these yield explicit closed forms for the corresponding subsequence sums: $\sum b(3n)q^n$, $\sum a(3n+1)q^n$, and $\sum \mu(3n+1)q^n$, in terms of Dedekind-like products $f_k$. The results deepen the analytic framework for mock theta functions by linking Appell–Lerch sums, theta parametrizations, and modular-quotient identities, with potential implications for congruences and further identities.
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 $μ_2(q)$. 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.