A cubic counterpart of Jacobi's identity and the AGM

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 / 3 and b_n+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.