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

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.