In this paper, we establish several new modular equations of degree 9 using Ramanujan’s modular equations. We also establish several general formulas for explicit evaluations of h9,n, h0 9,n, l9,n and l 0 9,n. As an application, we establish some explicit evaluations for Ramanujan’s cubic continued fraction

Chandankumar, S.

Mahadeva Naika, M.S.

Sushan Bairy, K.

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See	discussions,	stats,	and	author	profiles	for	this	publication	at:	https://www.researchgate.net/publication/261493913On	some	explicit	evaluation	of	the	ratios	ofRamanujan's	theta-functionArticle	·	January	2014CITATIONS5READS933	authors:M.	S.	Mahadeva	NaikaBangalore	University,	Central	College	Campus97	PUBLICATIONS			464	CITATIONS			SEE	PROFILEK.	Sushan	BairyVijaya	College,	Bangalore27	PUBLICATIONS			126	CITATIONS			SEE	PROFILEChandankumar	S.M.	S.	Ramaiah	University	of	Applied	Sciences29	PUBLICATIONS			110	CITATIONS			SEE	PROFILEAll	in-text	references	underlined	in	blue	are	linked	to	publications	on	ResearchGate,letting	you	access	and	read	them	immediately.Available	from:	Chandankumar	S.Retrieved	on:	21	September	2016ON SOME EXPLICIT EVALUATION OF THE RATIOS OF RAMANUJAN’STHETA–FUNCTIONM. S. MAHADEVA NAIKA, K. SUSHAN BAIRY AND S. CHANDANKUMARABSTRACT. In this paper, we establish several new modular equations of degree 9 usingRamanujan’s modular equations. We also establish several general formulas for explicitevaluations of h9,n, h′9,n, l9,n and l′9,n. As an application, we establish some explicitevaluations for Ramanujan’s cubic continued fraction.REFERENCES[1] C. Adiga, Taekyun Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramanujan’s cubic continuedfraction and explicit evaluations of theta-functions, Indian J. Pure 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 cubiccontinued fraction, The New Zealand J. Math., 31 (2002), 1–6.[3] N. D. Baruah and Nipen Saikia, Two parameters for Ramanujan’s theta–functions and their explicit values,Rocky Mountain J. Math., 37(6) (2007), 1747–1790.[4] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.[5] B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.[6] H. H. Chan, On Ramanujan’s cubic continued fraction, Acta Arith., 73 (1995), 343–355.[7] M. S. Mahadeva Naika, Some theorems on Ramanujan’s cubic continued fraction and related identities,Tamsui Oxf. J. Math. Sci. 24(3) (2008), 243–256.[8] M. S. Mahadeva Naika and S. Chandankumar, Some new modular equations and their applications, TamsuiOxf. J. Inf. Math. Sci., 28(4), (2013), 437–469.[9] M. S. Mahadeva Naika, M. C. Maheshkumar and K. Sushan Bairy, General formulas for explicit evaluationsof Ramanujan’s cubic continued fraction, Kyungpook Math. J., 49(3) (2009), 435–450.[10] M. S. Mahadeva Naika, K. Sushan Bairy and M. Manjunatha, Some new modular equations of degree fourand their explicit evaluations, Eur. J. Pure Appl. Math., 3(6) (2010), 924-947.[11] M. S. Mahadeva Naika, K. Sushan Bairy and S. Chandankumar, On some new modular equations of degreetwo akin to Ramanujan. Southeast Asian Bull. Math., (to appear).[12] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.[13] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.2000 Mathematics Subject Classification. 11B65, 11A55, 33D10, Secondary 11F27.Key words and phrases. Modular equation, Theta-function, Cubic continued fraction.Research supported by UGC major research project No.F.No.34–140\2008 (SR).12 M. S. MAHADEVA NAIKA, K. SUSHAN BAIRY AND S. CHANDANKUMAR[14] J. Yi, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math.Anal. Appl., 292 (2004), 381-400.[15] J. Yi, Yang Lee and Dae Hyun Paek, The explicit formulas and evaluations of Ramanujan’s theta-functionψ, J. Math. Anal. Appl., 321(2006), 157–181.(M. S. Mahadeva Naika) DEPARTMENT OF MATHEMATICS, BANGALORE UNIVERSITY, CENTRAL COL-LEGE CAMPUS, BANGALORE-560 001, INDIAE-mail address: msmnaika@rediffmail.com(K. Sushan Bairy) PG DEPARTMENT OF MATHEMATICS, VIJAYA COLLEGE, R.V. ROAD, BASAVANAGUDI,BANGALORE - 560 004E-mail address: ksbairy@rediffmail.com(S. Chandankumar) DEPARTMENT OF MATHEMATICS, ACHARYA INSTITUTE OF GRADUATE STUDIES,SOLDEVANAHALLI, HESARGHATTA MAIN ROAD, BENGALURU-560 107, KARNATAKA, INDIAE-mail address: chandan.s17@gmail.com

On some explicit evaluations of the ratios of Ramanujan's theta-function.

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On some explicit evaluations of the ratios of Ramanujan's theta-function.

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