Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T00:18:05.870Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE TENTH-ORDER MOCK THETA FUNCTIONS

Part of: Discontinuous groups and automorphic forms Sequences and sets

Published online by Cambridge University Press:  02 March 2017

ERIC T. MORTENSON
ERIC T. MORTENSON*
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany email [email protected]
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.

Using properties of Appell–Lerch functions, we give insightful proofs for six of Ramanujan’s identities for the tenth-order mock theta functions.

Keywords

mock theta functionsAppell–Lerch functions

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 1 , February 2018 , pp. 44 - 62
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Andrews, G. E., ‘Ramanujan’s fifth and seventh order mock theta functions’, Trans. Amer. Math. Soc. 293(1) (1986), 113–134.Google Scholar
Andrews, G. E. and Hickerson, D. R., ‘Ramanujan’s ‘lost’ notebook. VII: The sixth order mock theta functions’, Adv. Math. 89(1) (1991), 60–105.Google Scholar
Bringmann, K. and Ono, K., ‘Dyson’s ranks and Maass forms’, Ann. of Math. (2) 171 (2010), 419–449.Google Scholar
Bringmann, K., Ono, K. and Rhoades, R. C., ‘Eulerian series as modular forms’, J. Amer. Math. Soc. 21 (2008), 1085–1104.CrossRefGoogle Scholar
Choi, Y.-S., ‘Tenth order mock theta functions in Ramanujan’s lost notebook’, Invent. Math. 136(3) (1999), 497–569.Google Scholar
Choi, Y.-S., ‘Tenth order mock theta functions in Ramanujan’s lost notebook II’, Adv. Math. 156(2) (2000), 180–285.Google Scholar
Choi, Y.-S., ‘Tenth order mock theta functions in Ramanujan’s lost notebook III’, Proc. Lond. Math. Soc. (3) 94 (2007), 26–52.Google Scholar
Hickerson, D. R., ‘A proof of the mock theta conjectures’, Invent. Math. 94(3) (1988), 639–660.CrossRefGoogle Scholar
Hickerson, D. R. and Mortenson, E. T., ‘Hecke-type double sums, Appell–Lerch sums, and mock theta functions, I’, Proc. Lond. Math. Soc. (3) 109(2) (2014), 382–422.Google Scholar
Hickerson, D. R. and Mortenson, E. T., ‘Dyson’s ranks and Appell–Lerch sums’, Math. Ann. 367(1) (2017), 373–395.CrossRefGoogle Scholar
Koornwinder, T. H., ‘On the equivalence of two fundamental identities’, Anal. Appl. (Singap.) 12(6) (2014), 711–725.Google Scholar
Ramanujan, S., Collected Papers (Cambridge University Press, Cambridge, 1927), reprinted Chelsea, New York.Google Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988).Google Scholar
Watson, G. N., ‘The final problem: an account of the mock theta functions’, Proc. Lond. Math. Soc. (3) 42 (1937), 274–304.CrossRefGoogle Scholar
Weierstrass, K., ‘Zur Theorie der Jacobi’schen Funktionen von mehreren Veränderlichen’, Sitzungsber. Königl. Preuss. Akad. Wiss. (1882), 505–508; Werke, Vol. III, pp. 155–159 (1903).Google Scholar
Zwegers, S. P., ‘Mock theta functions’, PhD Thesis, Universiteit Utrecht, 2002.Google Scholar
Zwegers, S. P., ‘The tenth-order mock theta functions revisited’, Bull. Lond. Math. Soc. 42 (2010), 301–311.Google Scholar