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

Gauss’s arithmetic-geometric mean (AGM) which is described by two variables iteration $(a_n,b_n)\to (a_{n+1},b_{n+1})$ by $a_{n+1}=(a_n+b_n)\mathrm{/}2,b_{n+1}=\sqrt{a_n{b}_n}$. We extend it to three variables iteration $(a_n,b_n,c_n)\to (a_{n+1},b_{n+1},c_{n+1})$ 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 $a_{\mathrm{\infty }}=b_{\mathrm{\infty }}=M(a_0,b_0,c_0)$ and $c_{\mathrm{\infty }}=0$. The limit $M(a_0,b_0,c_0)$ is expressed by Appell’s hyper-geometric function $F_1(1\mathrm{/}2,\{1\mathrm{/}2,1\mathrm{/}2\},1;\kappa ,\lambda )$ 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.