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

Kiyoshi Sogo

paper

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

arXiv:2406.13077

Abstract

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 $a_{n + 1} = (a_{n} + b_{n}) /2, b_{n + 1} = a_{n} b_{n}$ . 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 $a_{\infty} = b_{\infty} = M (a_{0}, b_{0}, c_{0})$ and $c_{\infty} = 0$ . The limit $M (a_{0}, b_{0}, c_{0})$ is expressed by Appell's hyper-geometric function $F_{1} (1/2, {1/2, 1/2}, 1; κ, λ)$ of two variables $(κ, λ)$ 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.

10 pages, 0 figures