Title: An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme Show Chinese title

Title: 高斯算术-几何平均值（AGM）到三个变量迭代方案的扩展

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. Show Chinese abstract

Abstract: 高斯的算术-几何平均数（AGM），由两个变量迭代$(a_n, b_n)\rightarrow (a_{n+1}, b_{n+1})$和$a_{n+1}=(a_n+b_n)/2,\ b_{n+1}=\sqrt{a_nb_n}$描述。 我们将它扩展为三个变量迭代$(a_n, b_n, c_n)\rightarrow (a_{n+1}, b_{n+1}, c_{n+1})$，当$c_0=0$时退化为高斯的AGM。 我们从$a_0>b_0>c_0>0$开始迭代，并进一步限制$a_0>b_0+c_0$，收敛到$a_\infty=b_\infty=M(a_0, b_0, c_0)$和$c_\infty=0$。 极限$M(a_0, b_0, c_0)$由雅可比的超几何函数$F_1(1/2, \{1/2, 1/2\}, 1; \kappa, \lambda)$表示，该函数为两个变量$(\kappa, \lambda)$的函数，这些变量由$(a_0, b_0, c_0)$确定。 两种超几何函数（高斯的和雅可比的）之间的关系作为副产品被发现。