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Zbl 0725.33014
Borwein, J.M.; Borwein, P.B.
A cubic counterpart of Jacobi's identity and the AGM. (English)
[J] Trans. Am. Math. Soc. 323, No.2, 691-701 (1991). ISSN 0002-9947; ISSN 1088-6850/e

The authors' introduction: ``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 $$ a\sb{n+1}:=\frac{a\sb n+2b\sb n}{3}\text{ and } b\sb{n+1}:=\sp 3\sqrt{b\sb n(\frac{a\sp 2\sb n+a\sb nb\sb n+b\sp 2\sb n}{3})}. $$ The limit of this iteration is identified in terms of the hypergeometric function ${}\sb 2F(1/3,2/3;1;\cdot)$, which supports a particularly simple cubic transformation.
[J.Matkowski (Bielsko-Biała)]

MSC 2000:: *33E05 Elliptic functions and integrals
33C05 Classical hypergeometric functions
39B12 Iteraterative functional equations
11F27 Theta series; Weil representation

Keywords: Jacobi theta function; arithmetic-geometric mean iteration

Cited in: Zbl 1257.11043 Zbl 1118.11023 Zbl 1140.11358 Zbl 1110.11300 Zbl 0969.17017 Zbl 0949.33002 Zbl 0930.11046 Zbl 0906.33001 Zbl 0746.39006

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