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

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 :=a n +2b n 3andb n+1 := 3 b n (a n 2 +a n b n +b n 2 3)·

The limit of this iteration is identified in terms of the hypergeometric function 2 F(1/3,2/3;1;·), which supports a particularly simple cubic transformation.

33E05Elliptic functions and integrals
33C05Classical hypergeometric functions, 2 F 1
39B12Iterative and composite functional equations
11F27Theta series; Weil representation; theta correspondences