On Certain Theta Function Identities Analogous to Ramanujan’s P-Q Eta Function Identities
Kaliyur R. Vasuki, Abdulrawf Abdulrahman Abdullah Kahtan
.
DOI: 10.4236/am.2011.27117   PDF    HTML     6,533 Downloads   11,953 Views   Citations

Abstract

The purpose of this paper is to provide direct proofs of certain theta function identities analogous to Ramanujan’s P-Q eta functions identities.

Share and Cite:

K. Vasuki and A. Kahtan, "On Certain Theta Function Identities Analogous to Ramanujan’s P-Q Eta Function Identities," Applied Mathematics, Vol. 2 No. 7, 2011, pp. 874-882. doi: 10.4236/am.2011.27117.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] B. C. Berndt, “Ramanujan’s Notebooks, Part IV,” Sprin-ger-Verlag, New York, 1994.
[2] S. Ramanujan, “Notebooks (2 Volumes),” Tata Institute of Fundamental Research, Bombay, 1957.
[3] B. C. Berndt and L.-C. Zhang, “Ramanujan’s Identities for Eta Functions,” Mathematische Annalen, Vol. 292, No. 1, 1992, pp. 561-573. doi:10.1007/BF01444636
[4] S. Ramanujan, “The Lost Notebook and Other Unpublished Paper,” Narosa, New Delhi, 1988.
[5] B. C. Berndt, “Modular Equations in Ramanujan’s Lost Notebook,” In: R. P. Bamvah, V. C. Dumir and R. S. Hans-Gill, Eds., Number Theory, Hindustan Book Co., Delhi, 1999, pp. 55-74.
[6] B. C. Berndt and H. H. Chan, “Some Values for Rogers-Ramanujan Continued Fraction,” Canadian Journal of Mathematics, Vol. 47, 1995, pp. 897-914. doi:10.4153/CJM-1995-046-5
[7] B. C. Berndt, H. H. Chan and L.-C. Zhang, “Ramanujan’s Class Invariants and Cubic Continued Fraction,” Acta Arithmetica, Vol. 73, 1995, pp. 67-85.
[8] C. Adiga, K. R. Vasuki and M. S. M. Naika, “Some New Explicit Evaluations of Ramanujan’s Cubic Continued Fraction,” New Zealand Journal of Mathematics, Vol. 31, 2002, pp. 109-114.
[9] C. Adiga, K. R. Vasuki and K. Shivashankara, “Some Theta Function Identities and New Explicit Evaluation of Rogers-Ramanujan Continued Fraction,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 18, No. 1, 2002, pp. 101-117.
[10] N. D. Baruah, “On Some of Ramanujan’s Identities for Eta Functions,” Indian Journal of Mathematics, Vol. 43, 2000, pp. 253-266.
[11] N. D. Baruah and N. Saikia, “Some General Theorems on the Explicit Evaluations of Ramanujan’s Cubic Continued Fraction,” Journal of Computational and Applied Mathematics, Vol. 160, No. 1-2, 2003, pp. 37-51. doi:10.1016/S0377-0427(03)00612-5
[12] S. Bhargava, K. R. Vasuki and T. G. Sreeramamurthy, “Some Evaluations of Ramanujan’s Cubic Continued Fraction,” Indian Journal of Pure and Applied Mathematics, Vol. 35, 2004, pp. 1003-1025.
[13] S.-Y. Kang, “Ramanujan’s Formulas for Explicit Evaluation of the Rogers-Ramanujan Continued Fraction and Theta Functions,” Acta Arithmetica, Vol. 90, 1999, pp. 49-68.
[14] M. S. M. Naika and B. N. Dharmendra, “On Some New General Theorem for the Explicit Evaluations of Ramanujan’s Remarkable Product of Theta Function,” Ramanujan Journal, Vol. 15, No. 3, 2008, pp. 349-366. doi:10.1007/s11139-007-9081-1
[15] K. R. Vasuki, “On Some Ramanujan’s P-Q Modular Equations,” Journal of the Indian Mathematical Society, Vol. 73, No. 3-4, 2006, pp. 131-143.
[16] K. R. Vasuki and K. Shivashankara, “A Note on Explicit Evaluations of Products and Ratios of Class Invariants,” Math Forum, Vol. 13, 2000, pp. 45-46.
[17] K. R. Vasuki and T. G. Sreeramamurthy, “A Note on Explicit Evaluations of Ramanujan’s Continued Fraction,” Advanced Studies in Contemporary Mathematics, Vol. 9, 2004, pp. 63-80.
[18] K. R. Vasuki and B. R. Srivasta Kumar, “Certian Identities for Ramanujan-G¨ollnitz-Gordan Continued Fraction,” Journal of Computational and Applied Mathematics, Vol. 187, No. 1, 2006, pp. 87-95. doi:10.1016/j.cam.2005.03.038
[19] K. R. Vasuki and B. R. S. Kumar, “Evaluations of the Ramanujan-G¨ollnitz-Gordon Continued Fraction H(q) by Modular Equations,” Indian Journal of Mathematics, Vol. 48, No. 3, 2006, pp. 275-300.
[20] J. Yi, “Evaluations of the Rogers-Ramanujan’s Continued Fraction R(Q) by Modular Equations,” Acta Arithmetica, Vol. 97, No. 2, 2001, pp. 103-127. doi:10.4064/aa97-2-2
[21] C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, “Chapter 16 of Ramanujan’s Second Notebook: Theta Functions and Q-Series,” Memoirs of the American Mathematical Society, Vol. 53, No. 315, 1985, pp. 55-74.
[22] B. C. Berndt, “Ramanujan’s Notebooks, Part III,” Springer-Verlag, New York, 1991.
[23] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,” 4th Edition, Cambridge University Press, Cambridge, 1966.
[24] N. D. Baruah and R. Barman, “Certain Theta Function Identities and Ramanujan’s Modular Equations of Degree 3,” Indian Journal of Mathematics, Vol. 48, No. 3, 2006, pp. 113-133.
[25] K. R. Vasuki, G. Sharath and K. R. Rajanna, “Two Modular Equations for Squares of the Cubic-Functions with Applications,” Note di Mathematica (In Press).
[26] S.-Y. Kang, “Some Theorems in Rogers-Ramanujan Continued Fraction and Associated Theta Functions in Ramanujan’s Lost Notebook,” Ramanujan Journal, Vol. 3, No. 1, 1999, pp. 91-111. doi:10.1023/A:1009869426750
[27] S. Bhargava, K. R. Vasuki and K. R. Rajanna, “On Certain Identities of Ramanujan for Ratios of Eta Functions,” Preprint.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.