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A Cubic Counterpart of Jacobi's Identity and the AGM

J. M. Borwein and P. B. Borwein
Transactions of the American Mathematical Society
Vol. 323, No. 2 (Feb., 1991), pp. 691-701
DOI: 10.2307/2001551
Stable URL: http://www.jstor.org/stable/2001551
Page Count: 11
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A Cubic Counterpart of Jacobi's Identity and the AGM
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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 $$a_{n+1}: = \frac{a_n + 2b_n}{3} \text \,{and} b_{n+1}: = \root 3 \of {b_n \Bigg(\frac{a^2_n + a_nb_n + b^2_n}{3}\Bigg)}$$. 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.

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