Skip to main content Accessibility help
×
Home
Hostname: page-component-568f69f84b-jtg5s Total loading time: 0.207 Render date: 2021-09-18T08:57:36.349Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

ON SOME NEW MOCK THETA FUNCTIONS

Published online by Cambridge University Press:  21 December 2018

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
*Corresponding

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.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 454490.CrossRefGoogle Scholar
Andrews, G. E., ‘Ramanujan’s ‘Lost’ Notebook. I. Partial 𝜃-functions’, Adv. Math. 41 (1981), 137172.CrossRefGoogle Scholar
Andrews, G. E., Mordell Integrals and Ramanujan’s ‘lost’ Notebook, Lecture Notes in Mathematics, 899 (Springer, Berlin, 1981), 1048.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), 60105.CrossRefGoogle Scholar
Berndt, B. C. and Chan, S. H., ‘Sixth order mock theta functions’, Adv. Math. 216 (2007), 771786.CrossRefGoogle Scholar
Bringmann, K. and Ono, K., ‘The f (q) mock theta conjecture and partition ranks’, Invent. Math. 165 (2006), 243266.CrossRefGoogle Scholar
Bringmann, K. and Ono, K., ‘Dyson’s ranks and Maass forms’, Ann. of Math. (2) 171 (2010), 419449.CrossRefGoogle Scholar
Chen, B., ‘On the dual nature theory of bilateral series associated to mock theta functions’, Int. J. Number Theory 14 (2018), 6394.CrossRefGoogle Scholar
Dragonette, L., ‘Some asymptotic formulae for the mock theta series of Ramanujan’, Trans. Amer. Math. Soc. 72 (1952), 474500.CrossRefGoogle Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd edn (Cambridge University Press, Cambridge, 2004).CrossRefGoogle 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), 95144.CrossRefGoogle Scholar
Gu, N. S. S. and Liu, J., ‘Families of multisums as mock theta functions’, Adv. Appl. Math. 79 (2016), 98124.CrossRefGoogle Scholar
Hickerson, D. R., ‘A proof of the mock theta conjectures’, Invent. Math. 94 (1988), 639660.CrossRefGoogle Scholar
Hickerson, D. R., ‘On the seventh order mock theta functions’, Invent. Math. 94 (1988), 661677.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 (2014), 382422.CrossRefGoogle Scholar
Lovejoy, J., ‘Bailey pairs and indefinite quadratic forms’, J. Math. Anal. Appl. 410 (2014), 257273.CrossRefGoogle Scholar
Lovejoy, J. and Osburn, R., ‘The Bailey chain and mock theta functions’, Adv. Math. 238 (2013), 442458.CrossRefGoogle Scholar
Lovejoy, J. and Osburn, R., ‘ q-hypergeometric double sums as mock theta functions’, Pacific J. Math. 264 (2013), 151162.CrossRefGoogle Scholar
Lovejoy, J. and Osburn, R., ‘Mock theta double sums’, Glasg. Math. J. 59 (2017), 323348.CrossRefGoogle Scholar
McIntosh, R. J., ‘Second order mock theta functions’, Canad. Math. Bull. 50 (2007), 284290.CrossRefGoogle Scholar
McIntosh, R. J., ‘The H and K family of mock theta functions’, Canad. J. Math. 64 (2012), 935960.CrossRefGoogle Scholar
McIntosh, R. J., ‘New mock theta conjectures. Part I’, Ramanujan J. 46 (2018), 593604.CrossRefGoogle Scholar
Mortenson, E., ‘On the dual nature of partial theta functions and Appell–Lerch sums’, Adv. Math. 264 (2014), 236260.CrossRefGoogle 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), 347454.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), 315336.Google Scholar
Watson, G. N., ‘The final problem: an account of the mock theta functions’, J. Lond. Math. Soc. 11 (1936), 5580.CrossRefGoogle Scholar
Watson, G. N., ‘The mock theta functions (2)’, Proc. Lond. Math. Soc. (2) 42 (1937), 274304.CrossRefGoogle Scholar
Zagier, D., ‘Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann)’, Astérisque 326 (2009), 143164.Google Scholar
Zwegers, S. P., Mock theta functions, PhD Thesis, University of Utrecht, 2002.Google Scholar

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ON SOME NEW MOCK THETA FUNCTIONS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ON SOME NEW MOCK THETA FUNCTIONS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ON SOME NEW MOCK THETA FUNCTIONS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *