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 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.

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Acknowledgements

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

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Correspondence to Jane Y. X. Yang.

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The second named author was supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800604), and Doctor Scientific Research Foundation of Chongqing University of Posts and Telecommunications (Grant No. A2017-123).

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Li, F.Z.K., Yang, J.Y.X. Combinatorial proofs for identities related to generalizations of the mock theta functions \(\omega (q)\) and \(\nu (q)\). Ramanujan J 50, 527–550 (2019). https://doi.org/10.1007/s11139-018-0094-8

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Keywords

  • Partitions
  • Bijections
  • Mock theta functions
  • Odd Ferrers graph

Mathematics Subject Classification

  • 05A17
  • 05A19