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


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions, I. Q. J. Math. 17, 64–80 (1966)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions, II. Q. J. Math. 17, 132–143 (1966)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Pub. Co., New York (1976). Reissued: Cambridge University Press, New York (1998)

  4. 4.

    Andrews, G.E.: Partitions, Durfee-symbols, and the Atkin–Garvan moments of ranks. Invent. Math. 169, 173–188 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Andrews, G.E.: The Bhargava–Adiga summation and partitions. J. Indian Math. Soc. 84, 151–160 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Andrews, G.E.: Integer partitions with even parts below odd parts and the mock theta functions. Ann. Comb. 22, 433–445 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part V. Springer, New York (2018)

    Google Scholar 

  8. 8.

    Andrews, G.E., Yee, A.J.: Some identities associated with mock theta functions \(\omega (q) \) and \(\nu (q)\). Ramanujan J. (2018).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Andrews, G.E., Dixit, A., Yee, A.J.: Partitions associated with the Ramanujan/Watson mock theta functions \(\omega (q)\), \(\nu (q)\) and \(\phi (q)\). Res. Number Theory 1, 1–25 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Berndt, B.C., Yee, A.J.: Combinatorial proofs of identities in Ramanujans lost notebook associated with the Rogers–Fine identity and false theta functions. Ann. Comb. 7, 409–423 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chern, S.: Combinatorial proof of an identity of Andrews–Yee. Ramanujan J. (online)

  12. 12.

    Choi, Y.-S.: The basic bilateral hypergeometric series and the mock theta functions. Ramanujan J. 24, 345–386 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  14. 14.

    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    Google Scholar 

  15. 15.

    Watson, G.N.: The final problem: an account of the mock theta functions. J. Lond. Math. Soc. 11, 55–80 (1936)

    MathSciNet  Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Jane Y. X. Yang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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

Mathematics Subject Classification

  • 05A17
  • 05A19