A cubic counterpart of Jacobi’s identity and the AGM

Abstract:

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 \[ {a_{n + 1}}: = \frac {{{a_n} + 2{b_n}}} {3}\quad {\text {and}}\quad {b_{n + 1}}: = \sqrt [3]{{{b_n}\left ( {\frac {{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right ).}}\] The limit of this iteration is identified in terms of the hypergeometric function ${}_2{F_1}(1/3,2/3;1; \cdot )$, which supports a particularly simple cubic transformation.