超幾何関数の変換公式の導出

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はじめに

　この記事ではKummerの二次変換公式
$_{2} F_{1} (\begin{matrix} 2 a, 2 b \\ a + b + \frac{1}{2} \end{matrix}; z) =_{2} F_{1} (\begin{matrix} a, b \\ a + b + \frac{1}{2} \end{matrix}; 4 z (1 - z))$
を筆頭とした超幾何関数の変換公式を発見的に、そして網羅的に導出していきます。
　ただ今回の記事はいつもとは違い、予め書いておいたPDFを配布する形となります。
　そのリンクはこちら $\to$ hypergeometric.pdf

　ここではPDFで紹介した変換公式をまとめておくだけにとどめておきます。

変換公式まとめ

一次変換

$\begin{array}{rcl} _{2} F_{1} (\begin{array}{c} a, b \\ c \end{array}; z) & = & (1 - z)^{c - a - b}_{2} F_{1} (\begin{array}{c} c - a, c - b \\ c \end{array}; z) \\ = & (1 - z)^{- a}_{2} F_{1} (\begin{array}{c} a, c - b \\ c \end{array}; \frac{z}{z - 1}) = (1 - z)^{- b}_{2} F_{1} (\begin{array}{c} c - a, b \\ c \end{array}; \frac{z}{z - 1}) \\ = & \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)}_{2} F_{1} (\begin{array}{c} a, b \\ a + b - c + 1 \end{array}; 1 - z) + \frac{Γ (c) Γ (a + b - c)}{Γ (a) Γ (b)} (1 - z)^{c - a - b}_{2} F_{1} (\begin{array}{c} c - a, c - b \\ c - a - b + 1 \end{array}; 1 - z) \\ = & \frac{Γ (c) Γ (b - a)}{Γ (b) Γ (c - a)} (- z)^{- a}_{2} F_{1} (\begin{array}{c} a, a - c + 1 \\ a - b + 1 \end{array}; \frac{1}{z}) + \frac{Γ (c) Γ (a - b)}{Γ (a) Γ (c - b)} (- z)^{- b}_{2} F_{1} (\begin{array}{c} b, b - c + 1 \\ b - a + 1 \end{array}; \frac{1}{z}) \end{array}$

二次変換

$\begin{aligned} F (a, b; \frac{a + b + 1}{2}; z) & = F (\frac{a}{2}, \frac{b}{2}; \frac{a + b + 1}{2}; 4 z (1 - z)) \\ F (a, b; 2 b; z) & = (1 - z)^{- a} F (\frac{a}{2}, \frac{a + 1}{2} - b; a - b + 1; \frac{- 4 z}{(1 - z)^{2}}) \\ F (a, b; 2 b; z) & = (1 - z)^{- \frac{a}{2}} F (\frac{a}{2}, b - \frac{a}{2}; b + \frac{1}{2}; \frac{z^{2}}{- 4 (1 - z)}) \\ F (a, b; \frac{a + b + 1}{2}; z) & = (1 - 2 z)^{- a} F (\frac{a}{2}, \frac{a + 1}{2}; \frac{a + b + 1}{2}; \frac{- 4 z (1 - z)}{(1 - 2 z)^{2}}) \\ F (a, b; a - b + 1; z) & = (1 + z)^{- a} F (\frac{a}{2}, \frac{a + 1}{2}; a - b + 1; \frac{4 z}{(1 + z)^{2}}) \\ F (a, b; 2 b; z) & = {(1 - \frac{z}{2})}^{- a} F (\frac{a}{2}, \frac{a + 1}{2}; b + \frac{1}{2}; \frac{z^{2}}{(2 - z)^{2}}) \end{aligned}$

三次変換

$\begin{aligned} F (3 a, a + \frac{1}{3}; 2 a + \frac{2}{3}; x) & = (1 + ω^{2} x)^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{2}{3}; \frac{3 ω (ω - 1) x (1 - x)}{(x + ω)^{3}}) \\ F (3 a, \frac{1}{3} - a; \frac{1}{2}; z) & = (1 - z)^{- a} F (a, \frac{1}{6} - a; \frac{1}{2}; \frac{(9 - 8 z)^{2} z}{27 (1 - z)}) \\ F (3 a, a + \frac{1}{6}; \frac{1}{2}; z) & = (1 - z)^{- 2 a} F (a, \frac{1}{6} - a; \frac{1}{2}; \frac{(9 - z)^{2} z}{- 27 (1 - z)^{2}}) \\ F (3 a, \frac{1}{3} - a; \frac{1}{2}; z) & = {(1 - \frac{4}{3} z)}^{- 3 a} F (a, a + \frac{1}{3}; \frac{1}{2}; \frac{(9 - 8 z)^{2} z}{(4 z - 3)^{3}}) \\ F (3 a, a + \frac{1}{6}; \frac{1}{2}; z) & = {(1 + \frac{z}{3})}^{- 3 a} F (a, a + \frac{1}{3}; \frac{1}{2}; \frac{(z - 9)^{2} z}{(z + 3)^{3}}) \\ F (3 a, \frac{1}{3} - a; 2 a + \frac{5}{6}; z) & = (1 - 4 z)^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{27 z}{(4 z - 1)^{3}}) \\ F (3 a, a + \frac{1}{6}; 4 a + \frac{2}{3}; z) & = {(1 - \frac{z}{4})}^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{27 z^{2}}{(4 - z)^{3}}) \\ F (3 a, 3 a + \frac{1}{2}; 2 a + \frac{5}{6}; z) & = {(1 - \frac{3}{4} z)}^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{- 27 z^{2} (1 - z)}{(3 z - 4)^{3}}) \\ F (3 a, 3 a + \frac{1}{2}; 2 a + \frac{5}{6}; z) & = (1 + 3 z)^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{27 z (1 - z)^{2}}{(3 z + 1)^{3}}) \\ F (3 a, \frac{1}{3} - a; 2 a + \frac{5}{6}; z) & = (1 + 8 z)^{- 2 a} (1 - z)^{- a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{27 z}{(1 + 8 z)^{2} (1 - z)}) \\ F (3 a, a + \frac{1}{6}; 4 a + \frac{2}{3}; z) & = {(1 + \frac{z}{8})}^{- 2 a} (1 - z)^{- a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 27 z^{2}}{(z + 8)^{2} (1 - z)}) \\ F (3 a, 3 a + \frac{1}{2}; 4 a + \frac{2}{3}; z) & = {(1 - \frac{9}{8} z)}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 27 z^{2} (1 - z)}{(9 z - 8)^{2}}) \\ F (3 a, 3 a + \frac{1}{2}; 2 a + \frac{5}{6}; z) & = (1 - 9 z)^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 27 z (1 - z)^{2}}{(1 - 9 z)^{2}}) \end{aligned}$

四次変換

$\begin{aligned} F (4 a, \frac{1}{2} - 2 a; \frac{2}{3}; z) & = (1 - z)^{- a} F (a, \frac{1}{6} - a; \frac{2}{3}; \frac{(8 - 9 z)^{3} z}{64 (1 - z)}) \\ F (4 a, 2 a + \frac{1}{6}; \frac{2}{3}; z) & = (1 - z)^{- 3 a} F (a, \frac{1}{6} - a; \frac{2}{3}; \frac{(8 + z)^{3} z}{- 64 (1 - z)^{3}}) \\ F (4 a, \frac{1}{2} - 2 a; \frac{2}{3}; z) & = {(\frac{8 - 36 z + 27 z^{2}}{8})}^{- 2 a} F (a, a + \frac{1}{2}; \frac{2}{3}; \frac{(9 z - 8)^{3} z}{(8 - 36 z + 27 z^{2})^{2}}) \\ F (4 a, 2 a + \frac{1}{6}; \frac{2}{3}; z) & = {(\frac{8 - 20 z - z^{2}}{8})}^{- 2 a} F (a, a + \frac{1}{2}; \frac{2}{3}; \frac{(8 + z)^{3} z}{(z^{2} - 20 z - 8)^{2}}) \\ F (4 a, \frac{1}{2} - 2 a; 2 a + \frac{5}{6}; z) & = (1 - 9 z)^{- 3 a} (1 - z)^{- a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{64 z}{(9 z - 1)^{3} (1 - z)}) \\ F (4 a, 2 a + \frac{1}{6}; 6 a + \frac{1}{2}; z) & = {(1 - \frac{z}{9})}^{- 3 a} (1 - z)^{- a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{64 z^{3}}{(z - 9)^{3} (1 - z)}) \\ F (4 a, 4 a + \frac{1}{3}; 6 a + \frac{1}{2}; z) & = {(1 - \frac{8}{9} z)}^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{64 z^{3} (1 - z)}{(9 - 8 z)^{3}}) \\ F (4 a, 4 a + \frac{1}{3}; 2 a + \frac{5}{6}; z) & = (1 + 8 z)^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{64 z (1 - z)^{3}}{(1 + 8 z)^{3}}) \\ F (4 a, \frac{1}{2} - 2 a; 2 a + \frac{5}{6}; z) & = (1 + 18 z - 27 z^{2})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{64 z}{(1 + 18 z - 27 z^{2})^{2}}) \\ F (4 a, 2 a + \frac{1}{6}; 6 a + \frac{1}{2}; z) & = {(\frac{27 - 18 - z^{2}}{27})}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{64 z^{3}}{(z^{2} + 18 z - 27)^{2}}) \\ F (4 a, 4 a + \frac{1}{3}; 6 a + \frac{1}{2}; z) & = {(\frac{27 - 36 z + 8 z^{2}}{27})}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 64 z^{3} (1 - z)}{(8 z^{2} - 36 z + 27)^{2}}) \\ F (4 a, 4 a + \frac{1}{3}; 2 a + \frac{5}{6}; z) & = (1 - 20 z - 8 z^{2})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 64 z (1 - z)^{3}}{(1 - 20 z - 8 z^{2})^{2}}) \\ F (4 a, 2 a + \frac{1}{4}; 2 a + \frac{3}{4}; z) & = (1 + z)^{- 4 a} F (a, a + \frac{1}{4}; 2 a + \frac{3}{4}; \frac{16 z (1 - z)^{2}}{(1 + z)^{4}}) \\ F (4 a, 2 a + \frac{1}{4}; 4 a + \frac{1}{2}; z) & = {(1 - \frac{z}{2})}^{- 4 a} F (a, a + \frac{1}{4}; 2 a + \frac{3}{4}; \frac{16 z^{2} (1 - z)}{(2 - z)^{4}}) \\ F (4 a, \frac{1}{2}; 2 a + \frac{3}{4}; z) & = (1 - 2 z)^{- 4 a} F (a, a + \frac{1}{4}; 2 a + \frac{3}{4}; \frac{- 16 z (1 - z)}{(2 z - 1)^{4}}) \\ F (4 a, 2 a + \frac{1}{4}; 2 a + \frac{3}{4}; z) & = (1 - 6 z + z^{2})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{3}{4}; \frac{- 16 z (1 - z)^{2}}{(z^{2} - 6 z + 1)^{2}}) \\ F (4 a, 2 a + \frac{1}{4}; 4 a + \frac{1}{2}; z) & = {(\frac{4 - 4 z - z^{2}}{4})}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{3}{4}; \frac{- 16 z^{2} (1 - z)}{(z^{2} + 4 z - 4)^{2}}) \\ F (4 a, \frac{1}{2}; 2 a + \frac{3}{4}; z) & = (1 + 4 z - z^{2})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{3}{4}; \frac{16 z (1 - z)}{(1 + 4 z - 4 z^{2})^{2}}) \end{aligned}$

六次変換

$\begin{array}{rcl} F (6 a, 2 a + \frac{1}{2}; 4 a + \frac{2}{3}; z) & = & (1 - z + z^{2})^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{27 z^{2} (1 - z)^{2}}{4 (z^{2} - z + 1)^{3}}) \\ F (6 a, \frac{2}{3} - a; 2 a + \frac{5}{6}; z) & = & (1 - 16 z + 16 z^{2})^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{108 z (1 - z)}{(1 - 16 z + 16 z^{2})^{3}}) \\ F (6 a, 4 a + \frac{1}{6}; 2 a + \frac{5}{6}; z) & = & {(\frac{16 - 16 z + z^{2}}{16})}^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{- 108 z^{4} (1 - z)}{(z^{2} - 16 z + 16)^{3}}) \\ F (6 a, 4 a + \frac{1}{6}; 8 a + \frac{1}{3}; z) & = & (1 + 14 z + 14 z^{2})^{- 3 a} F (a, a + \frac{1}{3}; 2 a + \frac{5}{6}; \frac{- 108 z (1 - z)^{4}}{(1 + 14 z + 14 z^{2})^{3}}) \\ F (6 a, 2 a + \frac{1}{2}; 4 a + \frac{2}{3}; z) & = & {(\frac{2 - 3 z - 3 z^{2} + 2 z^{3}}{2})}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 27 z^{2} (1 - z)^{2}}{(2 z^{3} - 3 z^{2} - 3 z + 2)^{2}}) \\ F (6 a, \frac{2}{3} - a; 2 a + \frac{5}{6}; z) & = & (1 + 30 z - 96 z^{2} + 64 z^{3})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 108 z (1 - z)}{(64 z^{3} - 96 z^{2} + 30 z + 1)^{2}}) \\ F (6 a, 4 a + \frac{1}{6}; 2 a + \frac{5}{6}; z) & = & {(\frac{64 - 96 z + 30 z^{2} + z^{3}}{64})}^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{- 108 z^{4} (1 - z)}{(z^{3} + 30 z^{2} - 96 z + 64)^{2}}) \\ F (6 a, 4 a + \frac{1}{6}; 8 a + \frac{1}{3}; z) & = & (1 - 33 z - 33 z^{2} + z^{3})^{- 2 a} F (a, a + \frac{1}{2}; 2 a + \frac{5}{6}; \frac{108 z (1 - z)^{4}}{(z^{3} - 33 z - 33 z + 1)^{2}}) \end{array}$

無理変換

$\begin{aligned} F (a, b; a + b + \frac{1}{2}; z) & = F (2 a, 2 b; a + b + \frac{1}{2}; \frac{1 - \sqrt{1 - z}}{2}) \\ F (a, a + \frac{1}{2}; c; z) & = (1 - z)^{- a} F (2 a, 2 (c - a) - 1; c; \frac{\sqrt{1 - z} - 1}{2 \sqrt{1 - z}}) \\ F (a, a + \frac{1}{2}; c; z) & = (1 + \sqrt{z})^{- 2 a} F (2 a, c - \frac{1}{2}; 2 c - 1; \frac{2 \sqrt{z}}{1 + \sqrt{z}}) \\ F (a, b; a + b + \frac{1}{2}; z) & = (\sqrt{1 - z} + \sqrt{- z})^{- 2 a} F (2 a, a + b; 2 (a + b); \frac{2 \sqrt{- z}}{\sqrt{1 - z} + \sqrt{- z}}) \\ F (a, b; a + b + \frac{1}{2}; z) & = {(\frac{1 + \sqrt{1 - z}}{2})}^{- 2 a} F (2 a, a - b + \frac{1}{2}; a + b + \frac{1}{2}; \frac{\sqrt{1 - z} - 1}{\sqrt{1 - z} + 1}) \\ F (a, a + \frac{1}{2}; c; z) & = {(\frac{1 + \sqrt{1 - z}}{2})}^{- 2 a} F (2 a, 2 a - c + 1; c; \frac{1 - \sqrt{1 - z}}{1 + \sqrt{1 - z}}) \\ F (a, b; 2 b; z) & = (1 - z)^{- \frac{a}{2}} F (a, 2 b - a; b + \frac{1}{2}; \frac{(1 - \sqrt{1 - z})^{2}}{- 4 \sqrt{1 - z}}) \\ F (a, b; \frac{a + b + 1}{2}; z) & = (\sqrt{1 - z} + \sqrt{- z})^{- 2 a} F (a, \frac{a + b}{2}; a + b; \frac{4 \sqrt{z (1 - z)}}{(\sqrt{1 - z} + \sqrt{- z})^{2}}) \\ F (a, b; a - b + 1; z) & = (1 + \sqrt{z})^{- 2 a} F (a, a - b + \frac{1}{2}; 2 (a - b) + 1; \frac{4 \sqrt{z}}{(1 + \sqrt{z})^{2}}) \\ F (a, b; 2 b; z) & = {(\frac{1 + \sqrt{1 - x}}{2})}^{- 2 a} F (a, a - b + \frac{1}{2}; b + \frac{1}{2}; {(\frac{1 - \sqrt{1 - x}}{1 + \sqrt{1 - x}})}^{2}) \end{aligned}$