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.

21 pages, 6 figures