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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 an+1 := an + 2bn / 3 and bn+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation

A cubic counterpart of Jacobi’s identity and the AGM