An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme
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| Format: | Preprint |
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2024
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| _version_ | 1866916294173917184 |
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| author | Sogo, Kiyoshi |
| author_facet | Sogo, Kiyoshi |
| contents | 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 $F_1(1/2, \{1/2, 1/2\}, 1; κ, λ)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_13077 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme Sogo, Kiyoshi Classical Analysis and ODEs Number Theory 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 $F_1(1/2, \{1/2, 1/2\}, 1; κ, λ)$ 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. |
| title | An extension of Gauss's arithmetic-geometric mean (AGM) to three variables iteration scheme |
| topic | Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2406.13077 |