Abstract
We prove some new theta-function identities for two continued fractions of Ramanujan which are analogous to those of Ramanujan–Göllnitz–Gordon continued fraction. Then these identities are used to prove new general theorems for the explicit evaluations of the continued fractions.
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Andrews G.E.: On q-difference equations for certain well-poised basic hypergeometric series. Q. J. Math. (Oxford) 19, 433–447 (1968)
Baruah, N.D.; Saikia, N.: Modular equations and explicit values of Ramanujan–Selberg continued fraction. Int. J. Math. Math. Sci. Article ID 54901, 2006, 1–15 (2006)
Baruah N.D., Saikia N.: Explicit evaluations of Ramanujan–Göllnitz–Gordon continued fraction. Monatsh. Math. 154, 271–288 (2008)
Baruah, N.D.; Bora, J.; Saikia, N.: Some new proofs of modular relations for the Göllnitz–Gordon functions. Ramanujan J. 15, 281–30 (2008)
Berndt B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991)
Berndt, B.C.: Flowers which we cannot yet see growing in Ramanujan’s garden of hypergeometric series, elliptic functions, and q′s, in Special Functions 2000: Current Perspective and Future Directions. In: Bustoz, J.; Ismail, M.E.H.; Suslov, S.K. (eds.), Kluwer, Dordrecht. pp. 61–85 (2001)
Chan H.H., Huang S.-S.: On the Ramanujan–Göllnitz–Gordon continued fraction. Ramanujan J. 1, 75–90 (1997)
Göllnitz H.: Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967)
Gordon B.: Some continued fractions of Rogers–Ramanujan type. Duke Math. J. 32, 741–748 (1965)
Ramanathan K.G.: Ramanujan’s continued fraction. Indian J. Pure Appl. Math. 16, 695–724 (1985)
Ramanujan S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957)
Saikia N.: On modular identities of Ramanujan–Göllnitz–Gordon continued fraction. Far East J. Math. Sci. 54(1), 65–79 (2011)
Saikia N.: A new parameter for Ramanujan’s theta-fucntions and explicit values. Arab. J. Math. Sci. 18, 105–119 (2012)
Saikia N.: Modular identities and explicit evaluations of a continued fraction of Ramanujan. Int. J. Math. Math. Sci. 2012(694251), 1–10 (2012)
Saikia N.: A new continued fraction of Ramanujan, its modular identities and explicit evaluations. Afr. Mat. 26, 407–417 (2015)
Saikia, N.: Some new explicit values of Ramanujan–Selberg continued fraction (Submitted)
Saikia N.: General theorem for explicit evaluations and reciprocity theorems for Ramanujan–Göllnitz–Gordon continued fraction. Kyungpook Math. J. 55, 983–996 (2015)
Vasuki K.R., Srivatsa Kumar B.R.: Certain identities for Ramanujan–Göllnitz Gordon continued fraction. J. Comput. Appl. Math. 187, 87–95 (2006)
Yuttanan B.: New properties for the Ramanujan–Göllnitz–Gordon continued fraction. Acta Arith. 151(3), 293–310 (2012)
Yi, J.: Construction and Application of Modular Equations. Ph. D. Thesis, University of Illinois at Urbana Champaign (2004)
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Saikia, N. New theta-function identities and general theorems for the explicit evaluations of Ramanujan’s continued fractions. Arab. J. Math. 5, 145–158 (2016). https://doi.org/10.1007/s40065-016-0149-x
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DOI: https://doi.org/10.1007/s40065-016-0149-x