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

  • Frank Z. K. Li
  • Jane Y. X. YangEmail author
Article
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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 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.

Keywords

Partitions Bijections Mock theta functions Odd Ferrers graph 

Mathematics Subject Classification

05A17 05A19 

Notes

Acknowledgements

The authors appreciate the referee for his/her helpful comments which improved the quality of this manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Combinatorics, LPMCNankai UniversityTianjinPeople’s Republic of China
  2. 2.School of ScienceChongqing University of Posts and TelecommunicationsChongqingPeople’s Republic of China

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