The Ramanujan Journal

, Volume 36, Issue 1–2, pp 267–296 | Cite as

Universal mock theta functions and two-variable Hecke–Rogers identities

  • F. G. GarvanEmail author


We obtain two-variable Hecke–Rogers identities for three universal mock theta functions. This implies that many of Ramanujan’s mock theta functions, including all the third-order functions, have a Hecke–Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the \(M2\)-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.


Mock theta functions Hecke–Rogers identities Spt-functions 

Mathematics Subject Classification

11F27 11F37 11P82 33D15 



I would like to thank Krishna Alladi, Kathrin Bringmann, Freeman Dyson, Mike Hirschhorn, Robert Osburn, Steve Milne, Eric Mortenson and Martin Raum for their comments and suggestions. In particular, I thank Steve Milne for earlier pointing out his bijective proof [38] of (4.15), and I thank Eric Mortenson for his detailed comments and the results (7.3)–(7.8). Finally, I thank Doron Zeilberger for inviting me to present the preliminary results of this paper in his Experimental Math Seminar on April 25, 2013. See for the online video. Since this paper was submitted Kathy Ji and Aviva Zhao (“The Bailey transform and Hecke-Rogers identities for the universal mock theta functions, arXiv:1406.4398) have given new proofs of Theorem 1.1 using conjugate Bailey pairs. They have also been able to extend these results to infinite families of identities.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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