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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. Garvan
Article

Abstract

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.

Keywords

Mock theta functions Hecke–Rogers identities Spt-functions 

Mathematics Subject Classification

11F27 11F37 11P82 33D15 

Notes

Acknowledgments

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 http://youtu.be/oz2mdkd5jX4 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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