A cubic counterpart of Jacobi’s identity and the AGM. (English) Zbl 0725.33014

Borwein, J. M.; Borwein, P. B.

Trans. Am. Math. Soc. 323, No. 2, 691-701 (1991).

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 \[ a_{n+1}:=\frac{a_ n+2b_ n}{3}\text{ and } b_{n+1}:=^ 3\sqrt{b_ n(\frac{a^ 2_ n+a_ nb_ n+b^ 2_ n}{3})}. \] The limit of this iteration is identified in terms of the hypergeometric function \({}_ 2F(1/3,2/3;1;\cdot)\), which supports a particularly simple cubic transformation.

Reviewer: J.Matkowski (Bielsko-Biała)

MSC:

33E05	Elliptic functions and integrals
33C05	Classical hypergeometric functions, \({}_2F_1\)
39B12	Iteration theory, iterative and composite equations
11F27	Theta series; Weil representation; theta correspondences

Keywords:

Jacobi theta function; arithmetic-geometric mean iteration