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), 691-701

MSC:
Primary 33C75; Secondary 11F11, 11Y60, 33C05

DOI:
https://doi.org/10.1090/S0002-9947-1991-1010408-0

MathSciNet review:
1010408

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Abstract | References | Similar Articles | Additional Information

Abstract: We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1010408-0

Keywords:
Mean iterations,
theta functions,
hypergeometric functions,
generalised elliptic functions,
cubic transformations,
pi,
Ramanujan

Article copyright:
© Copyright 1991
American Mathematical Society