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

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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.

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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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  1. Department of Mathematics, University of Florida, Gainesville, FL, 32611-8105, USA

    F. G. Garvan

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  1. F. G. Garvan
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Correspondence to F. G. Garvan.

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Dedicated to the memory of Basil Gordon.

The author was supported in part by a grant from the Simon’s Foundation (#318714 to F. G. Garvan).

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Garvan, F.G. Universal mock theta functions and two-variable Hecke–Rogers identities. Ramanujan J 36, 267–296 (2015). https://doi.org/10.1007/s11139-014-9624-1

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