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
$$F(a,b;c;z)={}_2F_1\l(\begin{array}{c}a,b\\c\end{array};z\r)
=\sum^\infty_{n=0}\farc{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}$$
の変換公式をただ眺めます。
一覧
\begin{eqnarray} \tag{1}F(a,b;a-b+1;z) &=&(1-z)^{-a}F\l(\frac a2,\farc{a+1}2-b;a-b+1;-\frac{4z}{(1-z)^2}\r) \\\tag{2}F\l(2a,2b;a+b+\frac12;z\r)&=&F\l(a,b;a+b+\frac12;4z(1-z)\r) \\\tag{3}省&&略 \\\tag{4}F(a,b;2b;z) &=&\l(1-\frac z2\r)^{-a}F\l(\frac a2,\frac{a+1}2;b+\frac12;\l(\frac z{2-z}\r)^2\r) \\\tag{5}F\l(a,b;2b;\frac{4z}{(1+z)^2}\r) &=&(1+z)^{2a}F\l(a,a-b+\frac12;b+\frac12;z^2\r) \\\tag{6}F\l(a,a+\frac12;b;z(2-z)\r) &=&\l(1-\frac z2\r)^{-2a}F\l(2a,2a-b+1;b;\farc z{2-z}\r) \\\tag{7}省&&略 \\\tag{8}省&&略 \\\tag{9}省&&略 \\\tag{10}F\l(a,b;a+b+\frac12;z\r) &=&F\l(2a,2b;a+b+\frac12;\frac{1-\sqrt{1-z}}2\r) \\\tag{11}F\l(a,b;a+b+\frac12;z\r) &=&\l(\frac{1+\sqrt{1-z}}2\r)^{-2a}F\l(2a,a-b+\frac12;a+b+\frac12;\frac{\sqrt{1-z}-1}{\sqrt{1-z}+1}\r) \\\tag{12}F\l(a,b;a+b+\frac12;-z\r) &=&\l(\sqrt{1+z}+\sqrt z\r)^{-2a}F\l(2a,a+b;2a+2b;2\sqrt z(\sqrt{1+z}-\sqrt z)\r) \\\tag{13}F\l(a,b;a+b-\frac12;z\r) &=&\frac1{\sqrt{1-z}}F\l(2a-1,2b-1;a+b-\frac12;\frac{1-\sqrt{1-z}}2\r) \\\tag{14}F\l(a,b;a+b-\frac12;z\r) &=&\farc1{\sqrt{1-z}}\l(\frac{1+\sqrt{1-z}}2\r)^{1-2a} F\l(2a-1,a-b+\frac12;a+b-\frac12;\frac{\sqrt{1-z}-1}{\sqrt{1-z}+1}\r) \\\tag{15}F\l(a,b;a+b-\frac12;-z\r) &=&\frac{(\sqrt{1+z}+\sqrt z)^{1-2a}}{\sqrt{1+z}} F\l(2a-1,a+b-1;2a+2b-2;2\sqrt z(\sqrt{1+z}-\sqrt z)\r) \\\tag{16}F\l(a,a+\farc12;c;z\r) &=&(1-z)^{-a}F\l(2a,2c-2a-1;c;\frac{1-\sqrt{1-z}}2\r) \\\tag{17}F\l(a,a+\frac12;c;z\r) &=&(1+\sqrt z)^{-2a}F\l(2a,c-\frac12;2c-1;\frac{2\sqrt z}{1+\sqrt z}\r) \\\tag{18}F\l(a,b;\frac{a+b+1}2;z\r) &=&F\l(\frac a2,\frac b2;\frac{a+b+1}2;4z(1-z)\r) \\\tag{19}F\l(a,b;\frac{a+b+1}2;z\r) &=&(1-2z)F\l(\frac{a+1}2,\frac{b+1}2;\frac{a+b+1}2;4z(1-z)\r) \\\tag{20}F\l(a,b;\frac{a+b+1}2;z\r) &=&(1-2z)^{-a}F\l(\frac a2,\frac{a+1}2;\frac{a+b+1}2;-\frac{4z(1-z)}{(1-2z)^2}\r) \\\tag{21}F\l(a,b;\farc{a+b+1}2;-z\r) &=&(\sqrt{1+z}+\sqrt z)^{-2a} F\l(a,\frac{a+b}2;a+b;\frac{4\sqrt{z(1+z)}}{(\sqrt{1+z}+\sqrt z)^2}\r) \\\tag{22}F\l(a,1-a;c;z\r) &=&(1-z)^{c-1}F\l(\frac{c-a}2,\frac{c-a+1}2;c;4z(1-z)\r) \\\tag{23}F\l(a,1-a;c;z\r) &=&(1-z)^{c-1}(1-2z)F\l(\frac{c+a}2,\frac{c-a+1}2;c;4z(1-z)\r) \\\tag{24}F\l(a,1-a;c;z\r) &=&(1-z)^{c-1}(1-2z)^{a-c}F\l(\frac{c-a}2,\frac{c-a+1}2;c;-\frac{4z(1-z)}{(1-2z)^2}\r) \\\tag{25}F\l(a,1-a;c;-z\r) &=&(1-z)^{c-1}(\sqrt{1+z}+\sqrt z)^{2-2a-2c} F\l(c+a-1,c-\frac12;2c-1;\frac{4\sqrt{z(1+z)}}{(\sqrt{1+z}+\sqrt z)^2}\r) \\\tag{26}F\l(a,b;2b;z\r) &=&(1-z)^{-\frac a2}F\l(\frac a2,b-\frac a2;b+\frac12;-\frac{z^2}{4(1-z)}\r) \\\tag{27}F\l(a,b;2b;z\r) &=&\l(1-\frac z2\r)(1-z)^{-\farc{a+1}2} F\l(\farc{a+1}2,b-\frac{a-1}2;b+\frac12;-\frac{z^2}{4(1-z)}\r) \\\tag{28}F\l(a,b;2b;z\r) &=&\l(1-\frac z2\r)^{-a}F\l(\frac a2,\frac{a+1}2;b+\frac12;\frac{z^2}{(2-z)^2}\r) \\\tag{29}F\l(a,b;2b;z\r) &=&\l(1-\frac z2\r)^{a-2b}(1-z)^{b-a} F\l(b-\frac a2,b-\farc{a-1}2;b+\frac12;\frac{z^2}{(2-z)^2}\r) \\\tag{30}F\l(a,b;2b;z\r) &=&(1-z)^{-\frac a2}F\l(a,2b-a;b+\frac12;-\frac{(1-\sqrt{1-z})^2}{4\sqrt{1-z}}\r) \\\tag{31}F\l(a,b;2b;z\r) &=&\l(\frac{1+\sqrt{1-z}}2\r)^{-2a} F\l(a,a-b+\farc12;b+\farc12;\l(\frac{1-\sqrt{1-z}}{1+\sqrt{1-z}}\r)^2\r) \\\tag{32}F\l(a,b;a-b+1;z\r) &=&(1-z)^{-a}F\l(\frac a2,\frac{a+1}2-b;a-b+1;-\frac{4z}{(1-z)^2}\r) \\\tag{33}F\l(a,b;a-b+1;z\r) &=&(1+z)(1-z)^{-a-1}F\l(\frac{a+1}2,\frac a2-b+1;a-b+1;-\frac{4z}{(1-z)^2}\r) \\\tag{34}F\l(a,b;a-b+1;z\r) &=&(1+z)^{-a}F\l(\farc a2,\frac{a+1}2;a-b+1;\frac{4z}{(1+z)^2}\r) \\\tag{35}F\l(a,b;a-b+1;z\r) &=&(1+z)^{2b-a-1}(1-z)^{1-2b}F\l(\farc{a+1}2-b,\frac a2-b+1;a-b+1;\frac{4z}{(1+z)^2}\r) \\\tag{36}F\l(a,b;a-b+1;z\r) &=&(1+\sqrt z)^{-2a}F\l(a,a-b+\frac12;2a-2b+1;\frac{4\sqrt z}{(1+\sqrt z)^2}\r) \\\tag{37}省&&略 \\\tag{38}省&&略 \\\tag{39}F\l(\farc32a,\frac{3a-1}2;a+\frac12;-\sqrt{z^3}\r) &=&(1+z)^{1-3a}F\l(a,a-\frac13;2a;\frac{2z(3+z^2)}{(1+z)^3}\r) \\\tag{40}F\l(3a,3a+\frac12;4a+\frac23;z\r) &=&\l(1-\frac98z\r)^{-2a}F\l(a,a+\frac12;a+\frac56;-\frac{27z^2(1-z)}{(8-9z)^2}\r) \\\tag{41}F\l(3a,a+\frac12;2a+\frac56;z\r) &=&(1-9z)^{-2a}F\l(a,a+\frac12;2a+\frac56;-27\frac{z(1-z)^2}{(1-9z)^2}\r) \\\tag{42}F\l(3a,a+\frac16;4a+\frac23;z\r) &=&\l(1-\frac z4\r)^{-3a}F\l(a,a+\frac13;2a+\frac56;\frac{27z^2}{(4-z)^3}\r) \\\tag{43}F\l(3a,\frac13-a;2a+\frac56;z\r) &=&(1-4z)^{-3a}F\l(a,a+\frac13;2a+\frac56;-\frac{27z}{(1-4z)^3}\r) \\\tag{44}F\l(3a,\frac13-a;\farc12;z\r) &=&(1-z)^{-a}F\l(a,\frac16-a;\frac12;\frac{z(9-8z)^2}{27(1-z)}\r) \\\tag{45}F\l(3a,a+\frac16;\frac12;z\r) &=&(1-z)^{-2a}F\l(a,\frac16-a;\farc12;-\frac{z(9-z)^2}{27(1-z)^2}\r) \\\tag{46}F\l(3a+\frac12,\farc56-a;\frac32;z\r) &=&\l(1-\frac89z\r)\l(1-\frac43z\r)^{-3a-\frac32} F\l(a+\frac12,a+\frac56;\frac32;-\frac{z(9-8z)^2}{(3-4z)^3}\r) \\\tag{47}F\l(3a+\frac12,a+\frac23;\frac32;z\r) &=&\l(1-\frac z9\r)\l(1+\frac z3\r)^{-3a-\frac32} F\l(a+\frac12,a+\frac56;\frac32;\frac{z(9-z)^2}{(3+z)^3}\r) \end{eqnarray}
以上です。