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On the theory of elliptic functions based on ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$

Author(s): Li-Chien Shen
Journal: Trans. Amer. Math. Soc. 357 (2005), 2043-2058.
MSC (2000): Primary 11L05
Posted: November 4, 2004
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Abstract: Based on properties of the hypergeometric series ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$ , we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.

References:

1.: J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and AGM, Trans. Amer. Math. Soc. 323(1991), 691-701. MR 91e:33012
2.: A. Erdelyi (Editor),'' Higher Transcendental Functions'', Vol. 1, McGraw-Hill, New York, 1953. MR 15:419i
3.: Li-Chien Shen, On an identity of Ramanujan based on the hypergeometric series ${}_{2}F_{1}(1/3,2/3;1/2;x),$ J. Number Theory 69(1998), 125-134. MR 99d:11042
4.: Li-Chien Shen, On the modular equations of degree 3, Proc. Amer. Math. Soc. 122(1994), 1101-1114. MR 95b:11044
5.: E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.


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