Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T05:04:53.005Z Has data issue: false hasContentIssue false
  • English
  • Français

MOCK THETA DOUBLE SUMS

Part of: Basic hypergeometric functions Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 June 2016

JEREMY LOVEJOY  and
ROBERT OSBURN
JEREMY LOVEJOY
Affiliation:
CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France e-mail: lovejoy@math.cnrs.fr
ROBERT OSBURN
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland IHÉS, Le Bois-Marie, 35, route de Chartres, F-91440 Bures-sur-Yvette, France e-mails: robert.osburn@ucd.ie, osburn@ihes.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 323 - 348
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Andrews, G.E., Multiple series Rogers-Ramanujan identities, Pacific J. Math. 114 (1984), 267–283.CrossRefGoogle Scholar
2. Andrews, G.E., q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, in Regional Conference Series in Mathematics, vol. 66 (American Mathematical Society, Providence, RI, 1986).Google Scholar
3. Andrews, G.E., The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986), 113–134.Google Scholar
4. Andrews, G.E., Dyson, F. and Hickerson, D., Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391–407.Google Scholar
5. Andrews, G.E. and Hickerson, D., Ramanujan's “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105.Google Scholar
6. Andrews, G.E., Bailey pairs with free parameters, mock theta functions and tubular partitions, Ann. Combin. 18 (2014), 563–578.Google Scholar
7. Berkovich, A. and Warnaar, S.O., Positivity preserving transformations for q-binomial coefficients, Trans. Amer. Math. Soc. 357 (2005), 2291–2351.CrossRefGoogle Scholar
8. Bressoud, D., Ismail, M.E.H. and Stanton, D., Change of base in Bailey pairs, Ramanujan J. 4 (2000), 435–453.Google Scholar
9. Bringmann, K. and Kane, B., Multiplicative q-hypergeometric series arising from real quadratic fields, Trans. Amer. Math. Soc. 363 (4) (2011), 2191–2209.CrossRefGoogle Scholar
10. Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edition. Encyclopedia of Mathematics and its Applications, 96 (Cambridge University Press, Cambridge, 2004).Google Scholar
11. Gordon, B. and McIntosh, R.J., A survey of classical mock theta functions, in Partitions, q-series, and Modular Forms, Developments in Mathematics, vol. 23 (Springer, New York, 2012), 95–144.CrossRefGoogle Scholar
12. Hickerson, D. and Mortenson, E., Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. London Math. Soc. 109 (2014), 382–422.Google Scholar
13. Lovejoy, J. and Osburn, R., q-hypergeometric double sums as mock theta functions, Pacific J. Math. 264 (2013), no. 1, 151–162.Google Scholar
14. Lovejoy, J. and Osburn, R., Real quadratic double sums, Indag. Math. 26 (4) (2015), 697–712.CrossRefGoogle Scholar
15. Ono, K., Unearthing the visions of a master: harmonic Maass forms and number theory, in Proceedings of the 2008 Harvard-MIT Current Developments in Mathematics Conference (International Press, Somerville, MA, 2009), 347–454.Google Scholar
16. Slater, L.J., A new proof of Rogers's transformtions of infinite series, Proc. London Math. Soc. 53 (2) (1951), 460–475.CrossRefGoogle Scholar
17. Warnaar, S. O., Algebraic combinatorics and applications, in 50 years of Bailey's lemma, (Gößweinstein, 1999) (Springer, Berlin, 2001), 333–347.Google Scholar
18. Zagier, D., Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque 326 (2009), 143–164.Google Scholar
19. Zwegers, S., Mock Theta Functions, PhD Thesis (Universiteit Utrecht, 2002).Google Scholar