2024/06/18 by Kiyoshi Sogo, Sogo, Kiyoshi
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2406.13077
openalex publication_date 2024/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Gauss's arithmetic-geometric mean (AGM) which is described by two variables iteration (an, bn)→ (an+1, bn+1) by an+1=(an+bn)/2, bn+1=√(anbn). We extend it to three variables iteration (an, bn, cn)→ (an+1, bn+1, cn+1) which reduces to Gauss's AGM when c0=0. Our iteration starting from a0>b0>c0>0 with further restriction a0>b0+c0 converges to a_∞=b_∞=M(a0, b0, c0) and c_∞=0. The limit M(a0, b0, c0) is expressed by Appell's hyper-geometric function F1(1/2, \1/2, 1/2\, 1; κ, λ) of two variables (κ, λ) which are determined by (a0, b0, c0). A relation between two hyper-geometric functions (Gauss's and Appell's) is found as a by-product.