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ON SOME NEW MOCK THETA FUNCTIONS

Part of: Discontinuous groups and automorphic forms Basic hypergeometric functions

Published online by Cambridge University Press:  21 December 2018

NANCY S. S. GU  and
LI-JUN HAO Open the ORCID record for LI-JUN HAO [Opens in a new window]
NANCY S. S. GU
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China email gu@nankai.edu.cn
LI-JUN HAO*
Affiliation:
Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, PR China email haolijun152@163.com
*
✉haolijun152@163.com
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Abstract

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

Keywords

mock theta functionsBailey pairsAppell–Lerch sumspartial theta functions

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11F27: Theta series; Weil representation; theta correspondences
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 1 , August 2019 , pp. 53 - 66
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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Footnotes

The authors were supported by the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities of China.

References

Andrews, G. E., ‘On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions’, Amer. J. Math. 88 (1966), 454–490.Google Scholar
Andrews, G. E., ‘Ramanujan’s ‘Lost’ Notebook. I. Partial 𝜃-functions’, Adv. Math. 41 (1981), 137–172.Google Scholar
Andrews, G. E., Mordell Integrals and Ramanujan’s ‘lost’ Notebook, Lecture Notes in Mathematics, 899 (Springer, Berlin, 1981), 10–48.Google Scholar
Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook, Part I (Springer, New York, 2005).Google Scholar
Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook, Part II (Springer, New York, 2009).Google Scholar
Andrews, G. E. and Hickerson, D. R., ‘Ramanujan’s ‘lost’ notebook VII: the sixth order mock theta functions’, Adv. Math. 89 (1991), 60–105.Google Scholar
Berndt, B. C. and Chan, S. H., ‘Sixth order mock theta functions’, Adv. Math. 216 (2007), 771–786.Google Scholar
Bringmann, K. and Ono, K., ‘The f (q) mock theta conjecture and partition ranks’, Invent. Math. 165 (2006), 243–266.Google Scholar
Bringmann, K. and Ono, K., ‘Dyson’s ranks and Maass forms’, Ann. of Math. (2) 171 (2010), 419–449.Google Scholar
Chen, B., ‘On the dual nature theory of bilateral series associated to mock theta functions’, Int. J. Number Theory 14 (2018), 63–94.Google Scholar
Dragonette, L., ‘Some asymptotic formulae for the mock theta series of Ramanujan’, Trans. Amer. Math. Soc. 72 (1952), 474–500.Google Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edn (Cambridge University Press, Cambridge, 2004).Google Scholar
Gordon, B. and McIntosh, R. J., ‘A survey of classical mock theta functions’, in: Partitions, q-Series, and Modular Forms, Developments in Mathematics, 23 (Springer, New York, 2012), 95–144.Google Scholar
Gu, N. S. S. and Liu, J., ‘Families of multisums as mock theta functions’, Adv. Appl. Math. 79 (2016), 98–124.Google Scholar
Hickerson, D. R., ‘A proof of the mock theta conjectures’, Invent. Math. 94 (1988), 639–660.Google Scholar
Hickerson, D. R., ‘On the seventh order mock theta functions’, Invent. Math. 94 (1988), 661–677.Google 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 (2014), 382–422.Google Scholar
Lovejoy, J., ‘Bailey pairs and indefinite quadratic forms’, J. Math. Anal. Appl. 410 (2014), 257–273.Google Scholar
Lovejoy, J. and Osburn, R., ‘The Bailey chain and mock theta functions’, Adv. Math. 238 (2013), 442–458.Google Scholar
Lovejoy, J. and Osburn, R., ‘ q-hypergeometric double sums as mock theta functions’, Pacific J. Math. 264 (2013), 151–162.Google Scholar
Lovejoy, J. and Osburn, R., ‘Mock theta double sums’, Glasg. Math. J. 59 (2017), 323–348.Google Scholar
McIntosh, R. J., ‘Second order mock theta functions’, Canad. Math. Bull. 50 (2007), 284–290.Google Scholar
McIntosh, R. J., ‘The H and K family of mock theta functions’, Canad. J. Math. 64 (2012), 935–960.Google Scholar
McIntosh, R. J., ‘New mock theta conjectures. Part I’, Ramanujan J. 46 (2018), 593–604.Google Scholar
Mortenson, E., ‘On the dual nature of partial theta functions and Appell–Lerch sums’, Adv. Math. 264 (2014), 236–260.Google Scholar
Ono, K., The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, 102 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Ono, K., ‘Unearthing the visions of a master: harmonic Maass forms and number theory’, in: Proceedings of the 2008 Harvard-MIT Current Development in Mathematics Conference (International Press, Somerville, MA, 2009), 347–454.Google Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988).Google Scholar
Rogers, L. J., ‘On two theorems of combinatory analysis and some allied identities’, Proc. Lond. Math. Soc. (2) 53 (1916), 315–336.Google Scholar
Watson, G. N., ‘The final problem: an account of the mock theta functions’, J. Lond. Math. Soc. 11 (1936), 55–80.Google Scholar
Watson, G. N., ‘The mock theta functions (2)’, Proc. Lond. Math. Soc. (2) 42 (1937), 274–304.Google Scholar
Zagier, D., ‘Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann)’, Astérisque 326 (2009), 143–164.Google Scholar
Zwegers, S. P., Mock theta functions, PhD Thesis, University of Utrecht, 2002.Google Scholar