Combinatorial proofs for identities related to generalizations of themock theta functions omega(q) and nu (q)

摘要

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 begged 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.