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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 1010408
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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

$\displaystyle {a_{n + 1}}: = \frac{{{a_n} + 2{b_n}}} {3}\quad {\text{and}}\quad... ...}: = \sqrt[3]{{{b_n}\left( {\frac{{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right).}}$

The limit of this iteration is identified in terms of the hypergeometric function $ {}_2{F_1}(1/3,2/3;1; \cdot )$, which supports a particularly simple cubic transformation.

References [Enhancements On Off] (What's this?)

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Keywords: Mean iterations, theta functions, hypergeometric functions, generalised elliptic functions, cubic transformations, pi, Ramanujan
Article copyright: © Copyright 1991 American Mathematical Society