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On the reciprocity theorem of Ramanujan and its Generalizations

D. D. Somashekara|K. N. Murthy|S. L. Shalini
장전수학회 (Jangjeon Mathematical Society)|2012.07|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제15권 제3호|pp. 343-353|DOI 10.17777/pjms.2012.15.3.011

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참고문헌

키워드

q-seriesReciprocity theoremsJacobi’s triple product identityJackson’s 21 and 22- seriesCauchy parameter augmentation operator
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초록

In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we give an unified approach to the proofs of all the three reciprocity theorems. Some new q-gamma, q-beta and eta-function identities were also be deduced from the four variable reciprocity theorem.

Abstract

In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we give an unified approach to the proofs of all the three reciprocity theorems. Some new q-gamma, q-beta and eta-function identities were also be deduced from the four variable reciprocity theorem.

참고문헌 (20)

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  3. C. Adiga, "A note on four variable reciprocity theorem", Int. Jour. Math. Math Sci (2009)[DOI]
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  5. G.E. Andrews, "Ramanujans Lost Notebook, Part II" (2009), Springer
  6. R. Askey, "The q-gamma and q-beta functions", Applicable Anal, 8, pp.125~141 (1978)
  7. B.C. Berndt, "A reciprocity theorem for certain q-series found in Ramanujans lost notebook", Ramanujan J, 13, pp.27~37 (2007)
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  9. W.Y.C. Chen, "The Cauchy operator for basic hypergeometric series", Advances in Applied Mathematics, 41, pp.177~196 (2008)
  10. G. Gasper, "Basic hypergeometric series, Encyclopedia of Mathematics" (2004), Cambridge University press
  11. P.S. Guruprasad, "A simple proof of Ramanujans reciprocity theorem", Proc. Jangjeon Math. Soc, 9, pp.121~124 (2006)
  12. S.-Y. Kang, "Generalizations of Ramanujans reciprocity theorem and their applications", J. London Math. Soc, 75, pp.18~34 (2007)
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  15. D.D. Somashekara, "An interesting generalization of Jacobis triple product identity", Far East J. Math. Sci, 9, pp.255~259 (2003)
  16. D. D. Somashekara, "On the three variable reciprocity theorem and its applications", Aust. Jour. Math. anal. appl. (AJMAA), 9(1), pp.1~9 (2012)
  17. D.D. Somashekara, "On A Two Variable reciprocity theorem Of Ramanujan Bull", Pure Appl. Math
  18. D.D. Somashekara, "Finite forms of reciprocity theorems (Submitted for publication)", ~
  19. J. Thomae, "Beitr¨age zur theorie der durch die Heinesche reihe", J. Reine Angew. Math, 70, pp.258~281 (1869)
  20. Z. Zhang, "An identity related to Ramanujans and its applications", Indian J. Pure Appl. Math. (to appear)

관련 논문AI추천

Six-variable generalization of reciprocity theorem found in Ramanujan's lost notebookS. N. FATHIMA, YUDHISTHIRA JAMUDULIA장전수학회 · Advanced Studies in Contemporary Mathematics2016 · 제26권 제4호On some applications of the new forms of the reciprocity theoremsD. D. Somashekara, S. L. Shalini장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제3호On some identities of Ramanujan found in his lost notebookD.D. Somashekara, D. Mamta장전수학회 · Advanced Studies in Contemporary Mathematics2008 · 제16권 제2호ON SOME PARTIAL THETA FUNCTION IDENTITIES OF RAMANUJAN FOUND IN HIS LOST NOTEBOOKD. D. Somashekara, H. I. Kwon, S. L. Shalini, T. Kim장전수학회 · Advanced Studies in Contemporary Mathematics2015 · 제25권 제1호A simple proof of Ramanujan's reciprocity theoremN. Pradeep, P. S. Guruprasad장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2006 · 제9권 제2호CERTAIN MODULAR RELATIONS FOR REMARKABLE PRODUCT OF THETA-FUNCTIONSK. Sushan Bairy, M. S. MAHADEVA NAIKA, N. P. SUMAN장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2014 · 제17권 제3호

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