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