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A new proof of two identities involving Ramanujan’s cubic continued fraction

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In this paper, we give a new proof of two identities involving Ramanujan’s cubic continued fraction. These identities are the key ingredients to an analog of Ramanujan’s “Most Beautiful Identity” discovered recently.

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References

  1. Adiga, C., Kim, T., Mahadeva Naika, M.S., Madhusudhan, H.S.: On Ramanujan’s cubic continued fraction andexplicit evaluations of theta-functions. Indian J. Pure Appl. Math. 35, 1047–1062 (2004)

    MATH  MathSciNet  Google Scholar 

  2. Ahlgren, S.: The partition function modulo composite integers M. Math. Ann. 318, 795–803 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahlgren, S., Ono, K.: Congruence properties of the partition function. Proc. Natl. Acad. Sci. USA 98, 12882–12884 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Andrews, G.E.: On q-difference equations for certain well-poised basic hypergeometric series. Q. J. Math. (Oxford) 19, 433–447 (1968)

    Article  MATH  Google Scholar 

  5. Andrews, G.E.: The Theory of Partitions. Encycl. Math. and Its Appl., vol. 2. Addison-Wesley, Reading (1976). G.-C. Rota (ed.), Reissued: Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  6. Andrews, G.E.: An introduction to Ramanujan’s “lost” notebook. Am. Math. Mon. 86, 191–224 (1979)

    Google Scholar 

  7. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  8. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005)

    Google Scholar 

  9. Andrews, G.E., Eriksson, K.: Integer Partitions. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  10. Baruah, N.D.: Modular equations for Ramanujan’s cubic continued fraction. J. Math. Anal. Appl. 268, 244–255 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Baruah, N.D., Saikia, N.: Some general theorems on the explicit evaluations of Ramanujan’s cubic continued fraction. J. Comput. Appl. Math. 160, 37–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Baruah, N.D., Saikia, N.: Two parameters for Ramanujan’s theta-functions and their explicit values. Rocky Mt. J. Math. 37, 1747–1790 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)

    MATH  Google Scholar 

  14. Berndt, B.C.: Number Theory in the Spirit of Ramanujan. Am. Math. Soc., Providence (2004)

    Google Scholar 

  15. Berndt, B.C., Chan, H.H.: Some values for the Rogers–Ramanujan continued fraction. Can. J. Math. 47, 897–914 (1995)

    MATH  MathSciNet  Google Scholar 

  16. Berndt, B.C., Chan, H.H., Zhang, L.-C.: Ramanujan’s class invariants and cubic continued fraction. Acta Arith. 73, 67–85 (1995)

    MATH  MathSciNet  Google Scholar 

  17. Bhargava, S., Vasuki, K.R., Sreeramamurthy, T.G.: Some evaluations of Ramanujan’s cubic continued fraction. Indian J. Pure Appl. Math. 35, 1003–1025 (2004)

    MATH  MathSciNet  Google Scholar 

  18. Chan, H.C.: Ramanujan’s cubic continued fraction and a generalization of his “most beautiful identity”. Int. J. Number Theory (to appear)

  19. Chan, H.C.: Ramanujan’s cubic continued fraction and Ramanujan type congruences for a certain partition function. Int. J. Number Theory (to appear)

  20. Chan, H.C.: Distribution of a certain partition function modulo powers of primes. Preprint

  21. Chan, H.C., Ebbing, S.: Factorization theorems for the Rogers-Ramanujan continued fraction in the lost notebook. Preprint

  22. Chan, H.H.: On Ramanujan’s cubic continued fraction. Acta Arith. 73, 343–355 (1995)

    MathSciNet  Google Scholar 

  23. Chan, H.H., Loo, K.P.: Ramanujan’s cubic continued fraction revisited. Acta Arith. 126, 305–313 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gordon, B.: Some identities in combinatorial analysis. Q. J. Math. (Oxford) 12(1), 285–290 (1961)

    Article  MATH  Google Scholar 

  25. Hirschhorn, M.D.: Ramanujan’s contributions to continued fractions. In: Abdi, W.H. (ed.) Toils and Triumphs of Srinivasa Ramanujan, the Man and the Mathematician, pp. 236–246. National Publishing House, Jaipur (1992)

    Google Scholar 

  26. Hirschhorn, M.D.: An identity of Ramanujan, and applications. In: q-Series from a Contemporary Perspective. Contemporary Mathematics, vol. 254, pp. 229–234. Am. Math. Soc., Providence (2000)

    Google Scholar 

  27. Hirschhorn, M.D.: Ramanujan’s “most beautiful identity”. Aust. Math. Soc. Gaz. 31, 259–262 (2005)

    MathSciNet  Google Scholar 

  28. Ono, K.: Distribution of the partition function modulo M. Ann. Math. 151, 293–307 (2000)

    Article  MATH  Google Scholar 

  29. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  30. Selberg, A.: Über einige arithmetische Identitäten. Avh. Norske Vidensk.-Akad. Oslo I. Mat.-Naturv. Kl. 8, 3–23 (1936)

    Google Scholar 

  31. Selberg, A.: Collected Papers, vol. 1. Springer, Berlin (1989)

    MATH  Google Scholar 

  32. Yi, J., Lee, Y., Paek, D.H.: The explicit formulas andevaluations of Ramanujan’s theta-function psi. J. Math. Anal. Appl. 321, 157–181 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhang, L.-C.: q-difference equations and Ramanujan–Selberg continued fractions. Acta Arith. 57, 307–355 (1991)

    MATH  MathSciNet  Google Scholar 

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Chan, HC. A new proof of two identities involving Ramanujan’s cubic continued fraction. Ramanujan J 21, 173–180 (2010). https://doi.org/10.1007/s11139-009-9203-z

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  • DOI: https://doi.org/10.1007/s11139-009-9203-z

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