Abstract
Ramanujan recorded four reciprocity formulas for the Rogers–Ramanujan continued fractions. Two reciprocity formulas each are also associated with the Ramanujan–Göllnitz–Gordon continued fractions and a level-13 analog of the Rogers–Ramanujan continued fractions. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients of the quadratic equations to important formulas obtained by Ramanujan, examine a pattern in the relation between a continued fraction and its parameters, and use the reciprocity formulas to obtain close approximations for all values of the continued fractions. We highlight patterns in the expressions for certain explicit values of the Rogers–Ramanujan continued fractions by expressing them in terms of the golden ratio. We extend the analysis to reciprocity formulas for Ramanujan’s cubic continued fraction and the Ramanujan–Selberg continued fraction.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adiga, C., Kim, T., Naika, M.S., Madhusudhan, H.S.: On Ramanujan’s cubic continued fraction and explicit evaluations of theta-functions. Indian J. Pure Appl. Math. 35(9), 1047–1062 (2005)
Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook: Part I. Springer, New York (2005)
Bagis, N.: The complete evaluation of Rogers–Ramanujan and other continued fractions with elliptic functions. arXiv:1008.1304 (2010)
Baruah, N.D., Saikia, N.: Explicit evaluations of Ramanujan-Göllnitz–Gordon continued fraction. Mon. für Math. 154(4), 271–288 (2008)
Berndt, B.C., Chan, H.H.: Some values for the Rogers–Ramanujan continued fraction. Can. J. Math. 47(5), 897–914 (1995)
Chan, H.H.: On Ramanujan’s cubic continued fraction. Acta Arith. 73, 343–355 (1995)
Chan, H.H., Huang, S.S.: Ramanujan–Göllnitz–Gordon continued fraction. Ramanujan J. 1(1), 75–90 (1997)
Cooper, S.: Ramanujan’s Theta Functions. Springer, New York (2017)
Cooper, S., Ye, D.: Explicit evaluations of a level 13 analogue of the Rogers–Ramanujan continued fraction. J. Number Theory 139, 91–111 (2014)
Hardy, G.H.: The Indian Mathematician Ramanujan. Am. Math. Mon. 44(3), 137–155 (1937)
Kang, S.Y.: Ramanujan’s formulas for the explicit evaluation of the Rogers–Ramanujan continued fraction and theta-functions. Acta Arith. 90(1), 49–68 (1999)
Ramanujan, S.: Manuscript Book 2 of Srinivasa Ramanujan. Tata Institute of Fundamental Research, Bombay (1957)
Saikia, N.: Two theta-function identities for the Ramanujan–Selberg continued fraction and applications. J. Number Theory 151, 46–53 (2015)
Watson, G.N.: Theorems stated by Ramanujan (IX): two continued fractions. J. Lond. Math. Soc. 1(3), 231–237 (1929)
Yi, J.: Evaluations of the Rogers–Ramanujan continued fraction \(R(q)\) by modular equations. Acta Arith. 97(2), 103–127 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kohli, R. Properties of reciprocity formulas for the Rogers–Ramanujan continued fractions. Ramanujan J 51, 501–517 (2020). https://doi.org/10.1007/s11139-019-00153-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00153-0
Keywords
- Rogers–Ramanujan continued fractions
- Ramanujan–Göllnitz–Gordon continued fractions
- Ramanujan’s cubic continued fractions
- Ramanujan–Selberg continued fraction
- Reciprocal formulas