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CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETIC-GEOMETRIC MEAN ITERATION

CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETIC-GEOMETRIC MEAN ITERATION,Frank Garvan

CUBIC MODULAR IDENTITIES OF RAMANUJAN, HYPERGEOMETRIC FUNCTIONS AND ANALOGUES OF THE ARITHMETIC-GEOMETRIC MEAN ITERATION   (Citations: 7)
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There are four values ofs for which the hypergeometric function 2F1( 1 2 s; 1 2 +s; 1; ) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions 3(q), 4(q) and is related to the arithmetic-geometric mean iteration of Gauss and Legendre. Analogues of the arithmetic-geometric mean are given for the remaining cases. The case s = 1 6 and its relationship to the work of Ramanu- jan is highlighted. The work presented includes various pieces of joint work with combinations of the following: B. Berndt, S. Bhargava, J. Borwein, P. Borwein and M. Hirschhorn.
Published in 1993.
Cumulative Annual
    • ...Elliptic Integral Inequalities, with Applications G. D. Anderson, S.-L...
    • ...The connection ofK.r/ to the arithmetic‐geometric mean and other mean values was explored in [BB2], [G], and [VV]...
    • ...195 196 G. D. Anderson, S.-L. Qiu, and M. K. Vamanamurthy...
    • ...These results are related to normal forms for quartics [Be]. 198 G. D. Anderson, S.-L...
    • ...(2.7) 200 G. D. Anderson, S.-L. Qiu, and M. K. Vamanamurthy...
    • ...‚ ; 202 G. D. Anderson, S.-L. Qiu, and M. K. Vamanamurthy...
    • ...Qiu, and M. K. Vamanamurthy while G 00 p .r/ may be written as...
    • ...which, by Theorem 1.11(4), is increasing from.0; 1/ onto.0;1/. Hence G is increasing and convex...
    • ...Hence G is increasing and convex. 204 G. D. Anderson, S.-L...
    • ...They are grateful to the referee for significant corrections. 206 G. D. Anderson, S.-L...

    G. D. Andersonet al. Elliptic Integral Inequalities, with Applications

    • ...to parametrize a cubic mean iteration whose limit is identied with the the hypergeometric function 2F1( 1 3; 2 3 ; 1; ). For a survey of recent related results see [9]...

    Frank G. Garvan. A Combinatorial Proof of the Farkas-Kra Theta Function Identities and ...

    • ...For related recent work in this area see Borwein and Borwein (1991), Borwein, Borwein and Garvan (1994), Borwein, Borwein and Garvan (1993), Hirschhorn, Borwein and Garvan (1993), Shen (1994), Shen (manuscript) and Garvan (1994)...
    • ...For more discussion see Borwein and Borwein (1987), and Garvan (1994)...

    FRANK G. GARVANy. Ramanujan's theories of elliptic functions to alternative bases | a sy...

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