Generalizations of the Andrews-Yee identities associated with the mock theta functions $\omega(q)$ and $\nu(q)$

TL;DR

This paper extends Andrews-Yee identities to trivariate mock theta functions, partially solving a longstanding problem and deriving new and known results, including connections to overpartition and weighted partition functions.

Contribution

It generalizes Andrews-Yee identities to three-variable cases, addressing a problem highlighted by Li and Yang, and introduces new relations involving overpartition and weighted partition functions.

Findings

Partial generalization of Andrews-Yee identities to trivariate mock theta functions.

Derivation of new relations between overpartition and weighted partition functions.

Connection to Kang's three-variable reciprocity theorem.

Abstract

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third order mock theta functions $\omega(q)$ and $\nu(q)$, thereby extending their earlier results with the second author. Generalizing the Andrews-Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang's three-variable reciprocity theorem. A relation between a new restricted overpartition function $p^{*}(n)$ and a weighted partition function $p_*(n)$ is obtained from one of the special cases of our second theorem.