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


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.


Partitions Bijections Mock theta functions Odd Ferrers graph 

Mathematics Subject Classification

05A17 05A19 



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


  1. 1.
    Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions, I. Q. J. Math. 17, 64–80 (1966)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: On basic hypergeometric series, mock theta functions, and partitions, II. Q. J. Math. 17, 132–143 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Pub. Co., New York (1976). Reissued: Cambridge University Press, New York (1998)Google Scholar
  4. 4.
    Andrews, G.E.: Partitions, Durfee-symbols, and the Atkin–Garvan moments of ranks. Invent. Math. 169, 173–188 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Andrews, G.E.: The Bhargava–Adiga summation and partitions. J. Indian Math. Soc. 84, 151–160 (2017)MathSciNetCrossRefGoogle 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)Google Scholar
  7. 7.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part V. Springer, New York (2018)CrossRefGoogle Scholar
  8. 8.
    Andrews, G.E., Yee, A.J.: Some identities associated with mock theta functions \(\omega (q) \) and \(\nu (q)\). Ramanujan J. (2018).
  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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chern, S.: Combinatorial proof of an identity of Andrews–Yee. Ramanujan J. (online)Google Scholar
  12. 12.
    Choi, Y.-S.: The basic bilateral hypergeometric series and the mock theta functions. Ramanujan J. 24, 345–386 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  14. 14.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)zbMATHGoogle Scholar
  15. 15.
    Watson, G.N.: The final problem: an account of the mock theta functions. J. Lond. Math. Soc. 11, 55–80 (1936)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations