An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme

Abstract

Gauss's arithmetic-geometric mean (AGM) which is described by two variables iteration $(a_n, b_n)\rightarrow (a_{n+1}, b_{n+1})$ by $a_{n+1}=(a_n+b_n)/2,\ b_{n+1}=\sqrt{a_nb_n}$. We extend it to three variables iteration $(a_n, b_n, c_n)\rightarrow (a_{n+1}, b_{n+1}, c_{n+1})$ which reduces to Gauss's AGM when $c_0=0$. Our iteration starting from $a_0>b_0>c_0>0$ with further restriction $a_0>b_0+c_0$ converges to $a_\infty=b_\infty=M(a_0, b_0, c_0)$ and $c_\infty=0$. The limit $M(a_0, b_0, c_0)$ is expressed by Appell's hyper-geometric function $F_1(1/2, \{1/2, 1/2\}, 1; \kappa, \lambda)$ of two variables $(\kappa, \lambda)$ which are determined by $(a_0, b_0, c_0)$. A relation between two hyper-geometric functions (Gauss's and Appell's) is found as a by-product.