General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction

- Journal title : Kyungpook mathematical journal
- Volume 49, Issue 3, 2009, pp.435-450
- Publisher : Department of Mathematics, Kyungpook National University
- DOI : 10.5666/KMJ.2009.49.3.435

Title & Authors

General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction

Naika, Megadahalli Sidda Naika Mahadeva; Maheshkumar, Mugur Chinna Swamy; Bairy, Kurady Sushan;

Naika, Megadahalli Sidda Naika Mahadeva; Maheshkumar, Mugur Chinna Swamy; Bairy, Kurady Sushan;

Abstract

On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V () and also establish some explicit evaluations using the values of remarkable product of theta-function.

Keywords

cubic continued fraction;modular equation;theta-function;

Language

English

Cited by

References

1.

C. Adiga, T. Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramanujan's cubic continued fraction and explicit evaluations of theta-function, Indian J. Pure and Appl. Math., 35(9)(2004), 1047-1062.

2.

C. Adiga, K. R. Vasuki, and M. S. Mahadeva Naika, Some new explicit evaluations of Ramanujan's cubic continued fraction, The New Zealand J. Math., 31(2002), 1-6.

3.

N. D. Baruah, Modular equations for Ramanujan's cubic continued fraction, J. Math. Anal. and Appl., 268(2002), 244-255.

4.

N. D. Baruah and Nipen Saikia, Some general theorems on the explicit evaluations of Ramanujan's cubic continued fraction, J. Compu. and Appl. Math., 160(2003), 37-51.

5.

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

6.

B. C. Berndt, H. H. Chan and L. -C. Zhang, Ramanujan's remarkable product of the theta-function, Proc. Edinburgh Math. Soc., 40(1997), 583-612.

7.

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

8.

M. S. Mahadeva Naika, P-Q eta-function identities and computation of Ramanujan-Webber class invariants, J. Indian Math. Soc., 70(1-4), (2003), 121-134.

9.

M. S. Mahadeva Naika, Some theorems on Ramanujan's cubic continued fraction and related identities, Tamsui Oxford J. Math. Sci., 24(3)(2008), 243-256.

10.

M. S. Mahadeva Naika and B. N. Dharmendra, On some new general theorems for the explicit evaluations of Ramanujan's remarkable product of theta-function, The Ramanujan J., 15(3)(2008), 349-366.

11.

M. S. Mahadeva Naika, B. N. Dharmendra and K. Shivashankara, On some new explicit evaluations of Ramanujan's remarkable product of theta-function, South East Asian J. Math. and Math. Sci., 5(1)(2006), 107-119.

12.

M. S. Mahadeva Naika and M. C. Maheshkumar, Explicit evaluations of Ramanujan's remarkable product of theta-function, Adv. Stud. Contemp. Math., 13(2)(2006), 235-254.

13.

M. S. Mahadeva Naika, M. C. Maheshkumar and K. Sushan Bairy, On some re- markable product of theta-function, Aust. J. Math. Anal. Appl., 5(1)(2008), Art. 13, 1-15.

14.

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

15.

S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.