A cubic counterpart of Jacobi's identity and the AGM
Authors:
J. M. Borwein and P. B. Borwein
Journal:
Trans. Amer. Math. Soc. 323 (1991), 691701
MSC:
Primary 33C75; Secondary 11F11, 11Y60, 33C05
MathSciNet review:
1010408
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Abstract: We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmeticgeometric mean iteration of Gauss and Legendre. The iteration in question is The limit of this iteration is identified in terms of the hypergeometric function , which supports a particularly simple cubic transformation.
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 , Quadratic mean iterations, (monograph in preparation).
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 , More Ramanujantype series for , Ramanujan Revisited, Academic Press, 1988, pp. 359374.
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 S. Ramanujan, Modular equations and approximations to , Quart. J. Math. 45 (1914), 350372.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110104080
PII:
S 00029947(1991)10104080
Keywords:
Mean iterations,
theta functions,
hypergeometric functions,
generalised elliptic functions,
cubic transformations,
pi,
Ramanujan
Article copyright:
© Copyright 1991 American Mathematical Society
