Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-11T14:43:21.849Z Has data issue: false hasContentIssue false
  • English
  • Français

Second Order Mock Theta Functions

Published online by Cambridge University Press:  20 November 2018

Richard J. McIntosh
Richard J. McIntosh*
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2 e-mail: mcintosh@math.uregina.ca
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.

In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right)$, where $\left| q \right|<1$. He called them mock theta functions, because as $q$ radially approaches any point ${{e}^{2\pi ir}}\left( r\,\text{rational} \right)$, there is a theta function ${{F}_{r}}\left( q \right)$ with $F\left( q \right)-{{F}_{r}}\left( q \right)=O\left( 1 \right)$. In this paper we establish the relationship between two families of mock theta functions.

Keywords

11B6533D15q-seriesmock theta functionMordell integral
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 2 , 01 June 2007 , pp. 284 - 290
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Andrews, G. E., Mordell integrals and Ramanujan's “lost” notebook. Lecture Notes in Math. 899, Springer, Berlin, 1981, pp. 10–48.Google Scholar
[2] Apostol, T. M., Mathematical Analysis. Second edition, Addison-Wesley, Reading, MA, 1974.Google Scholar
[3] Gasper, G. and Rahman, M., Basic Hypergeometric Series. Cambridge University Press, Cambridge, 1990.Google Scholar
[4] Gordon, B. and McIntosh, R. J., Some eighth order mock theta functions. J. London Math. Soc. (2) 62(2000) 321–335.Google Scholar
[5] Ramanujan, S., Collected Papers. Cambridge University Press, 1927, reprinted by Chelsea, New York, 1962.Google Scholar
[6] Ramanujan, S., The Lost Notebook and Other Unpublished Papers. Springer-Verlag, Berlin, 1988.Google Scholar
[7] Rogers, L. J., On two theorems of combinatory analysis and some allied identities. Proc. London Math. Soc. (2) 16(1917), 315–336.Google Scholar
[8] Watson, G. N., The final problem: An account of the mock theta functions. J. London Math. Soc. 11(1936) 55–80.Google Scholar