超幾何関数の変換公式を眺める

はじめに

この記事では Higher transcendental functions の2.11 Quadratic and higher transformationsで列挙されている40余りの超幾何関数
$$F(a,b;c;z)={}_2{F}_1(\begin{array}{c}a,b\\ c\end{array};z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a{)}_n(b{)}_n}{(c{)}_n}\frac{z^n}{n!}$$
の変換公式をただ眺めます。

一覧

$$\begin{array}{}\text{(1)}& F(a,b;a-b+1;z)& =& (1-z{)}^{-a}F(\frac{a}{2},\frac{a+1}{2}-b;a-b+1;-\frac{4z}{(1-z{)}^2})\text{(2)}& F(2a,2b;a+b+\frac{1}{2};z)& =& F(a,b;a+b+\frac{1}{2};4z(1-z))\text{(3)}& 省& & 略\text{(4)}& F(a,b;2b;z)& =& {(1-\frac{z}{2})}^{-a}F(\frac{a}{2},\frac{a+1}{2};b+\frac{1}{2};{\left(\frac{z}{2-z}\right)}^2)\text{(5)}& F(a,b;2b;\frac{4z}{(1+z{)}^2})& =& (1+z{)}^{2a}F(a,a-b+\frac{1}{2};b+\frac{1}{2};z^2)\text{(6)}& F(a,a+\frac{1}{2};b;z(2-z))& =& {(1-\frac{z}{2})}^{-2a}F(2a,2a-b+1;b;\frac{z}{2-z})\text{(7)}& 省& & 略\text{(8)}& 省& & 略\text{(9)}& 省& & 略\text{(10)}& F(a,b;a+b+\frac{1}{2};z)& =& F(2a,2b;a+b+\frac{1}{2};\frac{1-\sqrt{1-z}}{2})\text{(11)}& F(a,b;a+b+\frac{1}{2};z)& =& {\left(\frac{1+\sqrt{1-z}}{2}\right)}^{-2a}F(2a,a-b+\frac{1}{2};a+b+\frac{1}{2};\frac{\sqrt{1-z}-1}{\sqrt{1-z}+1})\text{(12)}& F(a,b;a+b+\frac{1}{2};-z)& =& {(\sqrt{1+z}+\sqrt{z})}^{-2a}F(2a,a+b;2a+2b;2\sqrt{z}(\sqrt{1+z}-\sqrt{z}))\text{(13)}& F(a,b;a+b-\frac{1}{2};z)& =& \frac{1}{\sqrt{1-z}}F(2a-1,2b-1;a+b-\frac{1}{2};\frac{1-\sqrt{1-z}}{2})\text{(14)}& F(a,b;a+b-\frac{1}{2};z)& =& \frac{1}{\sqrt{1-z}}{\left(\frac{1+\sqrt{1-z}}{2}\right)}^{1-2a}F(2a-1,a-b+\frac{1}{2};a+b-\frac{1}{2};\frac{\sqrt{1-z}-1}{\sqrt{1-z}+1})\text{(15)}& F(a,b;a+b-\frac{1}{2};-z)& =& \frac{(\sqrt{1+z}+\sqrt{z}{)}^{1-2a}}{\sqrt{1+z}}F(2a-1,a+b-1;2a+2b-2;2\sqrt{z}(\sqrt{1+z}-\sqrt{z}))\text{(16)}& F(a,a+\frac{1}{2};c;z)& =& (1-z{)}^{-a}F(2a,2c-2a-1;c;\frac{1-\sqrt{1-z}}{2})\text{(17)}& F(a,a+\frac{1}{2};c;z)& =& (1+\sqrt{z}{)}^{-2a}F(2a,c-\frac{1}{2};2c-1;\frac{2\sqrt{z}}{1+\sqrt{z}})\text{(18)}& F(a,b;\frac{a+b+1}{2};z)& =& F(\frac{a}{2},\frac{b}{2};\frac{a+b+1}{2};4z(1-z))\text{(19)}& F(a,b;\frac{a+b+1}{2};z)& =& (1-2z)F(\frac{a+1}{2},\frac{b+1}{2};\frac{a+b+1}{2};4z(1-z))\text{(20)}& F(a,b;\frac{a+b+1}{2};z)& =& (1-2z{)}^{-a}F(\frac{a}{2},\frac{a+1}{2};\frac{a+b+1}{2};-\frac{4z(1-z)}{(1-2z{)}^2})\text{(21)}& F(a,b;\frac{a+b+1}{2};-z)& =& (\sqrt{1+z}+\sqrt{z}{)}^{-2a}F(a,\frac{a+b}{2};a+b;\frac{4\sqrt{z(1+z)}}{(\sqrt{1+z}+\sqrt{z}{)}^2})\text{(22)}& F(a,1-a;c;z)& =& (1-z{)}^{c-1}F(\frac{c-a}{2},\frac{c-a+1}{2};c;4z(1-z))\text{(23)}& F(a,1-a;c;z)& =& (1-z{)}^{c-1}(1-2z)F(\frac{c+a}{2},\frac{c-a+1}{2};c;4z(1-z))\text{(24)}& F(a,1-a;c;z)& =& (1-z{)}^{c-1}(1-2z{)}^{a-c}F(\frac{c-a}{2},\frac{c-a+1}{2};c;-\frac{4z(1-z)}{(1-2z{)}^2})\text{(25)}& F(a,1-a;c;-z)& =& (1-z{)}^{c-1}(\sqrt{1+z}+\sqrt{z}{)}^{2-2a-2c}F(c+a-1,c-\frac{1}{2};2c-1;\frac{4\sqrt{z(1+z)}}{(\sqrt{1+z}+\sqrt{z}{)}^2})\text{(26)}& F(a,b;2b;z)& =& (1-z{)}^{-\frac{a}{2}}F(\frac{a}{2},b-\frac{a}{2};b+\frac{1}{2};-\frac{z^2}{4(1-z)})\text{(27)}& F(a,b;2b;z)& =& (1-\frac{z}{2})(1-z{)}^{-\frac{a+1}{2}}F(\frac{a+1}{2},b-\frac{a-1}{2};b+\frac{1}{2};-\frac{z^2}{4(1-z)})\text{(28)}& F(a,b;2b;z)& =& {(1-\frac{z}{2})}^{-a}F(\frac{a}{2},\frac{a+1}{2};b+\frac{1}{2};\frac{z^2}{(2-z{)}^2})\text{(29)}& F(a,b;2b;z)& =& {(1-\frac{z}{2})}^{a-2b}(1-z{)}^{b-a}F(b-\frac{a}{2},b-\frac{a-1}{2};b+\frac{1}{2};\frac{z^2}{(2-z{)}^2})\text{(30)}& F(a,b;2b;z)& =& (1-z{)}^{-\frac{a}{2}}F(a,2b-a;b+\frac{1}{2};-\frac{(1-\sqrt{1-z}{)}^2}{4\sqrt{1-z}})\text{(31)}& F(a,b;2b;z)& =& {\left(\frac{1+\sqrt{1-z}}{2}\right)}^{-2a}F(a,a-b+\frac{1}{2};b+\frac{1}{2};{\left(\frac{1-\sqrt{1-z}}{1+\sqrt{1-z}}\right)}^2)\text{(32)}& F(a,b;a-b+1;z)& =& (1-z{)}^{-a}F(\frac{a}{2},\frac{a+1}{2}-b;a-b+1;-\frac{4z}{(1-z{)}^2})\text{(33)}& F(a,b;a-b+1;z)& =& (1+z)(1-z{)}^{-a-1}F(\frac{a+1}{2},\frac{a}{2}-b+1;a-b+1;-\frac{4z}{(1-z{)}^2})\text{(34)}& F(a,b;a-b+1;z)& =& (1+z{)}^{-a}F(\frac{a}{2},\frac{a+1}{2};a-b+1;\frac{4z}{(1+z{)}^2})\text{(35)}& F(a,b;a-b+1;z)& =& (1+z{)}^{2b-a-1}(1-z{)}^{1-2b}F(\frac{a+1}{2}-b,\frac{a}{2}-b+1;a-b+1;\frac{4z}{(1+z{)}^2})\text{(36)}& F(a,b;a-b+1;z)& =& (1+\sqrt{z}{)}^{-2a}F(a,a-b+\frac{1}{2};2a-2b+1;\frac{4\sqrt{z}}{(1+\sqrt{z}{)}^2})\text{(37)}& 省& & 略\text{(38)}& 省& & 略\text{(39)}& F(\frac{3}{2}a,\frac{3a-1}{2};a+\frac{1}{2};-\sqrt{z^3})& =& (1+z{)}^{1-3a}F(a,a-\frac{1}{3};2a;\frac{2z(3+z^2)}{(1+z{)}^3})\text{(40)}& F(3a,3a+\frac{1}{2};4a+\frac{2}{3};z)& =& {(1-\frac{9}{8}z)}^{-2a}F(a,a+\frac{1}{2};a+\frac{5}{6};-\frac{27z^2(1-z)}{(8-9z{)}^2})\text{(41)}& F(3a,a+\frac{1}{2};2a+\frac{5}{6};z)& =& (1-9z{)}^{-2a}F(a,a+\frac{1}{2};2a+\frac{5}{6};-27\frac{z(1-z{)}^2}{(1-9z{)}^2})\text{(42)}& F(3a,a+\frac{1}{6};4a+\frac{2}{3};z)& =& {(1-\frac{z}{4})}^{-3a}F(a,a+\frac{1}{3};2a+\frac{5}{6};\frac{27z^2}{(4-z{)}^3})\text{(43)}& F(3a,\frac{1}{3}-a;2a+\frac{5}{6};z)& =& (1-4z{)}^{-3a}F(a,a+\frac{1}{3};2a+\frac{5}{6};-\frac{27z}{(1-4z{)}^3})\text{(44)}& F(3a,\frac{1}{3}-a;\frac{1}{2};z)& =& (1-z{)}^{-a}F(a,\frac{1}{6}-a;\frac{1}{2};\frac{z(9-8z{)}^2}{27(1-z)})\text{(45)}& F(3a,a+\frac{1}{6};\frac{1}{2};z)& =& (1-z{)}^{-2a}F(a,\frac{1}{6}-a;\frac{1}{2};-\frac{z(9-z{)}^2}{27(1-z{)}^2})\text{(46)}& F(3a+\frac{1}{2},\frac{5}{6}-a;\frac{3}{2};z)& =& (1-\frac{8}{9}z){(1-\frac{4}{3}z)}^{-3a-\frac{3}{2}}F(a+\frac{1}{2},a+\frac{5}{6};\frac{3}{2};-\frac{z(9-8z{)}^2}{(3-4z{)}^3})\text{(47)}& F(3a+\frac{1}{2},a+\frac{2}{3};\frac{3}{2};z)& =& (1-\frac{z}{9}){(1+\frac{z}{3})}^{-3a-\frac{3}{2}}F(a+\frac{1}{2},a+\frac{5}{6};\frac{3}{2};\frac{z(9-z{)}^2}{(3+z{)}^3})\end{array}$$

以上です。