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arXiv · 1802.00959

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

Frank Z. K. Li·Jane Y. X. Yang
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Abstract

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

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Frank Z. K. Li, Jane Y. X. Yang. 2018-08-10. Combinatorial proofs for identities related to generalizations of the mock theta functions $ω(q)$ and $ν(q)$. https://doi.org/10.1007/s11139-018-0094-8

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