The Ramanujan Journal

, Volume 30, Issue 2, pp 279–308 | Cite as

On three third order mock theta functions and Hecke-type double sums

  • Eric Mortenson


We obtain four Hecke-type double sums for three of Ramanujan’s third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews’ work on q-orthogonal polynomials and Bringmann, Hikami, and Lovejoy’s work on unified Witten–Reshetikhin–Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.


Hecke-type double sums Appell–Lerch sums Mock theta functions Indefinite theta series 

Mathematics Subject Classification (2000)

11B65 11F11 11F27 


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  1. 1.
    Andrews, G.E.: Hecke modular forms and the Kac–Peterson identities. Trans. Am. Math. Soc. 283, 451–458 (1984) zbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The fifth and seventh order mock theta functions. Trans. Am. Math. Soc. 293, 113–134 (1986) zbMATHCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: q-orthogonal polynomials, Rogers–Ramanujan identities, and mock theta functions. Proc. Steklov Inst. Math. (to appear) Google Scholar
  4. 4.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook Part I. Springer, New York (2005) Google Scholar
  5. 5.
    Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. Lond. Math. Soc. 4(3), 84–106 (1954) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bringmann, K., Hikami, K., Lovejoy, J.: On the modularity of the unified WRT invariants of certain Seifert manifolds. Adv. Appl. Math. 46, 86–93 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hickerson, D.R.: A proof of the mock theta conjectures. Invent. Math. 94(3), 639–660 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hickerson, D.R.: On the seventh order mock theta functions. Invent. Math. 94(3), 661–677 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hickerson, D.R., Mortenson, E.T.: Hecke-type double sums, Appell–Lerch sums, and mock theta functions (I) (submitted) Google Scholar
  10. 10.
    Kac, V., Peterson, D.: Infinite-dimensional Lie algebras, theta functions and modular forms. Adv. Math. 53(2), 125–264 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lerch, M.: Poznámky k theorii funkcí elliptických. Rozpr. Čes. Akad. Císaře Františka Josefa Vědy Slovesn. Umění, v praze 24, 465–480 (1892) Google Scholar
  12. 12.
    Lerch, M.: Bemerkungen zur Theorie der elliptischen Funktionen. Jahrb. Fortschr. Math. 24, 442–445 (1892) Google Scholar
  13. 13.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa Publishing House, New Delhi (1988) zbMATHGoogle Scholar
  14. 14.
    Watson, G.N.: The final problem: an account of the mock theta functions. J. Lond. Math. Soc. 11, 55–80 (1936) CrossRefGoogle Scholar
  15. 15.
    Zwegers, S.P.: Mock theta functions. Ph.D. Thesis, Universiteit Utrecht (2002) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsThe University of QueenslandBrisbaneAustralia

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