Abstract
By employing a method of parameterizations for Ramanujan's theta-functions, we find several modular relations and explicit values of the Ramanujan-Selberg continued fractions.
Open Access
Modular relations and explicit values of Ramanujan-Selberg continued fractions
Received12 Sept 2005
Revised13 Apr 2006
Accepted25 Apr 2006
Published13 Aug 2006
References
B. C. Berndt, Ramanujan's Notebooks. Part III, Springer, New York, 1991.
View at: Zentralblatt MATH | MathSciNetB. C. Berndt, Ramanujan's Notebooks. Part V, Springer, New York, 1998.
View at: Zentralblatt MATH | MathSciNetK. G. Ramanathan, “Hypergeometric series and continued fractions,” Proceedings of the Indian Academy of Sciences. Mathematical Sciences, vol. 97, no. 1–3, pp. 277–296 (1988), 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957.
View at: Zentralblatt MATH | MathSciNetS. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
View at: Zentralblatt MATH | MathSciNetA. Selberg, “Über einige arithmetische Identitäten,” Avhandlinger utgitt av Det Norske Videnskaps-Akademi i Oslo. I. Matematisk-Naturvidenskapelig Klasse, no. 8, pp. 2–23, 1936.
View at: Google Scholar | Zentralblatt MATHK. R. Vasuki and K. Shivashankara, “On Ramanujan's continued fractions,” Ganita, vol. 53, no. 1, pp. 81–88, 2002.
View at: Google Scholar | MathSciNetE. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1966, Indian edition is published by Universal Book Stall, New Delhi, 1991.
J. Yi, Construction and application of modular equation, M.S. thesis, University of Illionis, Illionis, 2001.
J. Yi, “Theta-function identities and the explicit formulas for theta-function and their applications,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 381–400, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetL.-C. Zhang, “Explicit evaluations of a Ramanujan-Selberg continued fraction,” Proceedings of the American Mathematical Society, vol. 130, no. 1, pp. 9–14, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.