General Theorem for Explicit Evaluations and Reciprocity Theorems for Ramanujan-Göllnitz-Gordon Continued Fraction

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 4,  2015, pp.983-996
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.4.983
Title & Authors
General Theorem for Explicit Evaluations and Reciprocity Theorems for Ramanujan-Göllnitz-Gordon Continued Fraction
SAIKIA, NIPEN;

Abstract
In the paper A new parameter for Ramanujan's theta-functions and explicit values, Arab J. Math. Sc., 18 (2012), 105-119, Saikia studied the parameter $\small{A_{k,n}}$ involving Ramanujan's theta-functions $\small{{\phi}(q)}$ and $\small{{\psi}(q)}$ for any positive real numbers k and n and applied it to find explicit values of $\small{{\psi}(q)}$. As more application to the parameter $\small{A_{k,n}}$, in this paper we prove a new general theorem for explicit evaluation of Ramanujan-$\small{G{\ddot{o}}llnitz}$-Gordon continued fraction K(q) in terms of the parameter $\small{A_{k,n}}$ and give examples. We also find some new explicit values of the parameter $\small{A_{k,n}}$ and offer reciprocity theorems for the continued fraction K(q).
Keywords
Ramanujan's theta-functions;Ramanujan-$\small{G{\ddot{o}}llnitz}$-Gordon continued fraction;
Language
English
Cited by
1.
New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions, Arabian Journal of Mathematics, 2016, 5, 3, 145
References
1.
N. D. Baruah and N. Saikia, Explicit evaluations of Ramanujan-Gollnitz-Gordon continued fraction, Monatsh. Math., 154(2008), 271-288.

2.
B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York, 1991.

3.
H. H. Chan, On Ramanujan's cubic continued fraction, Acta Arith., 73(1995), 343-355.

4.
H. H. Chan and S.-S. Huang, On the Ramanujan-Gollnitz-Gordon continued fraction, Ramanujan J., 1(1997), 75-90.

5.
K. G. Ramanathan, On Ramanujans continued fraction, Acta Arith., 43(1984), 209-226.

6.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

7.
N. Saikia, A new parameter for Ramanujan's theta-fucntions and explicit values, Arab J. Math. Sc., 18(2012), 105-119.

8.
N. Saikia, On modular identities of Ramanujan-Gollnitz-Gordon continued fraction, Far East J. Math. Sc., 54(1)(2011), 65-79.

9.
K. R. Vasuki and B. R. Srivatsa Kumar, Certain identities for Ramanujan-Gollnitz Gordon continued fraction, J. Comput. Appl. Math., 187(2006), 87-95.

10.
G. N.Watson, Theorems stated by Ramanujan(ix): two continued fractions, J. London Math. Soc., 4(1929), 231237.

11.
E. 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.