Abstract
In both his second and lost notebooks, Ramanujan introduced and studied a function \(k(q)=r(q)r^2(q^2)\), where \(r(q)\) is the Rogers–Ramanujan continued fraction. Ramanujan also recorded five beautiful relations between the Rogers–Ramanujan continued fraction \(r(q)\) and the five continued fractions \(r(-q)\), \(r(q^2)\), \(r(q^3)\), \(r(q^4)\), and \(r(q^5)\). Motivated by those relations, we present some modular relations between \(k(q)\) and \(k(-q)\), \(k(-q^2)\), \(k(q^3)\), and \(k(q^5)\) in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Berndt, B.C.: The Continued Fractions Found in the Unorganized Portions of Ramanujan’s Notebook, vol. 477. Memoirs of the American Mathematical Society, Providence (1992)
Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part I. Springer, New York (2005)
Berndt, B.C.: Ramanujan’s Notebooks. Part III. Springer, New York (1991)
Berndt, B.C.: Ramanujan’s Notebooks. Part V. Springer, New York (1998)
Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)
Cooper, S.: On Ramanujan’s function \(k(q)=r(q)r^2(q^2)\). Ramanujan J. 20, 311–328 (2009)
Cooper, S., Hirschhorn, M.D.: Factorizations that involve Ramanujan’s function \(k(q)=r(q)r^2(q^2)\). Acta Math. Sin. (Engl. Ser.) 27, 2301–2308 (2011)
Gugg, C.: Two modular equations for squares of the Rogers–Ramanujan functions with applications. Ramanujan J. 18, 183–207 (2009)
Kang, S.Y.: Some theorems on the Rogers–Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook. Ramanujan J. 3, 91–111 (1999)
Raghavan, S., Rangachari, S.S.: Ramanujan’s elliptic integrals and modular identities. In: Number Theory and Related Topics, pp. 119–149. Oxford University Press, Bombay (1989)
Ramanathan, K.G.: On Ramanujan’s continued fraction. Acta Arith. 43, 209–226 (1984)
Ramanujan, S.: Notebooks, 2 vols. Tata Institute of Fundamental Research, Bombay (1957)
Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)
Rogers, L.J.: On a type of modular relation. Proc. Lond. Math. Soc. 19, 387–397 (1921)
Watson, G.N.: Theorems stated by Ramanujan (VII): theorems on continued fractions. J. Lond. Math. Soc. 4, 39–48 (1929)
Yi, J.: Modular equations for the Rogers–Ramanujan continued fraction and the Dedekind eta-function. Ramanujan J. 5, 377–384 (2001)
Acknowledgments
The authors are grateful to the anonymous referee for his/her valuable suggestions, corrections, and comments which resulted in a great improvement of the original manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (11201188).
Rights and permissions
About this article
Cite this article
Xia, E.X.W., Yao, O.X.M. Some modular relations for Ramanujan’s function \(k(q)=r(q)r^2(q^2)\) . Ramanujan J 35, 243–251 (2014). https://doi.org/10.1007/s11139-014-9576-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-014-9576-5