Combinatorial proofs for identities related to generalizations of the mock theta functions $\omega(q)$ and $\nu(q)$

TL;DR

This paper introduces combinatorial proofs for identities involving generalized mock theta functions related to partition functions, extending earlier results and providing bijective proofs for recent analytical identities.

Contribution

It defines new trivariate mock theta functions and offers combinatorial and bijective proofs for identities connecting these functions to partition functions.

Findings

Extended identities involving $oldsymbol{ ext{omega}(y,z;q)}$ and $oldsymbol{ ext{nu}(y,z;q)}$.

Provided bijective proofs for Andrews-Yee identities.

Connected combinatorial interpretations with generalized mock theta functions.

Abstract

The two partition functions $p_\omega(n)$ and $p_\nu(n)$ were introduced by Andrews, Dixit and Yee, which are related to the third order mock theta functions $\omega(q)$ and $\nu(q)$, respectively. Recently, Andrews and Yee analytically studied two identities that connect the refinements of $p_\omega(n)$ and $p_\nu(n)$ with the generalized bivariate mock theta functions $\omega(z;q)$ and $\nu(z;q)$, respectively. However, they stated these identities cried out for bijective proofs. In this paper, we first define the generalized trivariate mock theta functions $\omega(y,z;q)$ and $\nu(y,z;q)$. Then by utilizing odd Ferrers graph, we obtain certain identities concerning to $\omega(y,z;q)$ and $\nu(y,z;q)$, which extend some early results of Andrews that are related to $\omega(z;q)$ and $\nu(z;q)$. In virtue of the combinatorial interpretations that arise from the identities involving $\omega(y,z;q)$ and $\nu(y,z;q)$, we finally present bijective proofs for the two identities of Andrews-Yee.