Combinatorial proofs for identities related with generalizations of the mock theta functions $ω(q)$ and $ν(q)$

Frank Z. K. Li, Jane Y. X. Yang

Published 2018-02-03Version 1

The two partition functions $p_\omega(n)$ and $p_\nu(n)$ were introduced by Andrews, Dixit and Yee, which are related with 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 cried out for bijective proofs of these two identities. 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 with $\omega(y,z;q)$ and $\nu(y,z;q)$, which extend some early results of Andrews that are related with $\omega(z;q)$ 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.