Hecke-Rogers double-sums and false theta functions


We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.

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  1. 1.

    Andrews, G.E., Dyson, F.J., Hickerson, D.R.: Partitions and indefinite quadratic forms. Invent. Math. 91(3), 391–407 (1988)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Andrews, G.E., Hickerson, D.R.: Ramanujan’s ‘lost’ Notebook. VII: the sixth order mock theta functions. Adv. Math. 89, 60–105 (1991)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Andrews, G.E., Warnaar, O.: The Bailey transform and false theta functions. Ramanujan J. 14, 173–188 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Andrews, G.E.: Partitions with Distinct Evens, Advances in Combinatorial Mathematics, pp. 31–37. Springer, Berlin (2009)

    Book  Google Scholar 

  5. 5.

    Chan, S.H., Kim, B.: On some double-sum false theta series. J. Number Theory 190, 40–55 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chu, W., Zhang, W.: Bilateral Bailey lemma and false theta functions. Int. J. Number Theory 6(3), 515–577 (2010)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cohen, H.: \(q\)-Identities for Maass waveform. Invent. Math. 91, 409–422 (1988)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Corson, D., Favero, D., Liesinger, K., Zubairy, S.: Characters and \(q\)-series in \({\mathbb{Q}}(\sqrt{2})\). J. Number Theory 107, 392–405 (2004)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gordon, B., McIntosh, R.: Some eighth order mock theta functions. J. Lond. Math. Soc. (2) 62, 321–335 (2000)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hecke, E.: Über einen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen, pp. 418–427. Mathematische Werke, Vandenhoeck and Ruprecht, Göttingen (1959)

    Google Scholar 

  11. 11.

    Hickerson, D.R.: A proof of the mock theta conjectures. Invent. Math 94, 639–660 (1988)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hickerson, D.R.: On the seventh order mock theta functions. Invent. Math 94, 661–677 (1988)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hickerson, D.R., Mortenson, E.T.: Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I. Proc. Lond. Math. Soc. (3) 109(2), 382–422 (2014)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Polishchuk, A.: Indefinite theta series of signature (1,1) from the point of view of homological mirror symmetry. Adv. Math. 196(1), 1–51 (2005)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)

    MathSciNet  Google Scholar 

  16. 16.

    Warnaar, S.O.: 50 years of Bailey’s lemma. In: Betten, A., et al. (eds.) Algebraic Combinatorics and Applications, pp. 333–347. Springer, Berlin (2001)

    Google Scholar 

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This research was supported by Ministry of Science and Higher Education of the Russian Federation, agreement No. 075-15-2019-1619, and by the Theoretical Physics and Mathematics Advancement Foundation BASIS, agreement No. 20-7-1-25-1.

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Correspondence to Eric T. Mortenson.

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Mortenson, E.T. Hecke-Rogers double-sums and false theta functions. Res. number theory 7, 28 (2021). https://doi.org/10.1007/s40993-021-00256-y

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  • Hecke-Rogers double-sums
  • False theta functions
  • Mock theta functions

Mathematics Subject Classification

  • 11B65
  • 11F27