一般化超幾何関数の微分と積分
一般化超幾何関数の微分と積分
一般化超幾何関数\(F\left(\boldsymbol{a};\boldsymbol{b};x\right)\)の微分と積分は次のようになる。
微分
積分
一般化超幾何関数\(F\left(\boldsymbol{a};\boldsymbol{b};x\right)\)の微分と積分は次のようになる。
微分
(1)
\[ \frac{d}{dx}F\left(\boldsymbol{a};\boldsymbol{b};x\right)=\frac{\prod_{i=1}^{\dim\boldsymbol{a}}a_{i}}{\prod_{j=1}^{\dim\boldsymbol{b}}b_{j}}F\left(\boldsymbol{a}+\boldsymbol{1};\boldsymbol{b}+\boldsymbol{1};x\right) \](2)
\[ \frac{d}{dx}x^{u}F\left(\boldsymbol{a};\boldsymbol{b};cx^{v}\right)=ux^{u-1}F\left(\boldsymbol{a},\frac{u}{v}+1;\boldsymbol{b},\frac{u}{v};cx^{v}\right) \]積分
(3)
\[ \int F\left(\boldsymbol{a};\boldsymbol{b};x\right)dx=\frac{\prod_{j=1}^{\dim\boldsymbol{b}}\left(b_{j}-1\right)}{\prod_{i=1}^{\dim\boldsymbol{a}}\left(a_{i}-1\right)}F\left(\boldsymbol{a}-\boldsymbol{1};\boldsymbol{b}-\boldsymbol{1};x\right)+C \](4)
\[ \int x^{u}F\left(\boldsymbol{a};\boldsymbol{b};cx^{v}\right)dx=\frac{x^{u+1}}{u+1}F\left(\boldsymbol{a},\frac{u+1}{v};\boldsymbol{b},\frac{u+1}{v}+1;cx^{v}\right)+C \]\begin{align*}
F\left(\boldsymbol{a};\boldsymbol{b};x\right) & =F\left(a_{1},a_{2},\cdots,a_{\dim\boldsymbol{a}};b_{1},b_{2},\cdots,b_{\dim\boldsymbol{b}};x\right)
\end{align*}
\[
F\left(\boldsymbol{a+a_{0}};\boldsymbol{b+b_{0}};x\right)=F\left(a_{1}+a_{0},a_{2}+a_{0},\cdots,a_{\dim\boldsymbol{a}}+a_{0};b_{1}+b_{0},b_{2}+b_{0},\cdots,b_{\dim\boldsymbol{b}}+b_{0};x\right)
\]
で表している。
\[ \frac{d}{dx}F\left(a,b;c;x\right)=\frac{ab}{c}F\left(a+1,b+1;c+1;x\right) \] \[ \frac{d}{dx}x^{u}F\left(a,b;c;tx^{v}\right)=ux^{u-1}F\left(a,b,\frac{u}{v}+1;c,\frac{u}{v};tx^{v}\right) \] \[ \int F\left(a,b;c;x\right)dx=\frac{c-1}{\left(a-1\right)\left(b-1\right)}F\left(a-1,b-1;c-1;x\right)+C \] \[ \int x^{u}F\left(a,b;c;tx^{v}\right)dx=\frac{x^{u+1}}{u+1}F\left(a,b,\frac{u+1}{v};c,\frac{u+1}{v}+1;tx^{v}\right)+C \]
-
超幾何関数については次のようになります。\[ \frac{d}{dx}F\left(a,b;c;x\right)=\frac{ab}{c}F\left(a+1,b+1;c+1;x\right) \] \[ \frac{d}{dx}x^{u}F\left(a,b;c;tx^{v}\right)=ux^{u-1}F\left(a,b,\frac{u}{v}+1;c,\frac{u}{v};tx^{v}\right) \] \[ \int F\left(a,b;c;x\right)dx=\frac{c-1}{\left(a-1\right)\left(b-1\right)}F\left(a-1,b-1;c-1;x\right)+C \] \[ \int x^{u}F\left(a,b;c;tx^{v}\right)dx=\frac{x^{u+1}}{u+1}F\left(a,b,\frac{u+1}{v};c,\frac{u+1}{v}+1;tx^{v}\right)+C \]
(1)
\begin{align*} \frac{d}{dx}F\left(\boldsymbol{a};\boldsymbol{b};x\right) & =\frac{d}{dx}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{x^{k}}{k!}\\ & =\sum_{k=1}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{x^{k-1}}{\left(k-1\right)!}\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k+1\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k+1\right)}\frac{x^{k}}{k!}\\ & =\frac{\prod_{i=1}^{\dim\boldsymbol{a}}a_{i}}{\prod_{j=1}^{\dim\boldsymbol{b}}b_{j}}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i}+1,k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j}+1,k\right)}\frac{x^{k}}{k!}\\ & =\frac{\prod_{i=1}^{\dim\boldsymbol{a}}a_{i}}{\prod_{j=1}^{\dim\boldsymbol{b}}b_{j}}F\left(\boldsymbol{a}+\boldsymbol{1};\boldsymbol{b}+\boldsymbol{1};x\right) \end{align*}(2)
\begin{align*} \frac{d}{dx}x^{u}F\left(\boldsymbol{a};\boldsymbol{b};cx^{v}\right) & =\frac{d}{dx}x^{u}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{\left(cx^{v}\right)^{k}}{k!}\\ & =\frac{d}{dx}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{c^{k}x^{u+vk}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{\left(u+vk\right)c^{k}x^{u+vk-1}}{k!}\\ & =x^{u-1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{v\left(k+\frac{u}{v}\right)\left(cx^{v}\right)^{k}}{k!}\\ & =x^{u-1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{vQ\left(\frac{u}{v}+1,k\right)\left(cx^{v}\right)^{k}}{\frac{v}{u}Q\left(\frac{u}{v},k\right)k!}\\ & =ux^{u-1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{Q\left(\frac{u}{v}+1,k\right)\left(cx^{v}\right)^{k}}{Q\left(\frac{u}{v},k\right)k!}\\ & =ux^{u-1}F\left(\boldsymbol{a},\frac{u}{v}+1;\boldsymbol{b},\frac{u}{v};cx^{v}\right) \end{align*}(3)
\begin{align*} \int F\left(\boldsymbol{a};\boldsymbol{b};x\right)dx & =\int\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{x^{k}}{k!}dx\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{x^{k+1}}{\left(k+1\right)!}+C\\ & =\sum_{k=1}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k-1\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k-1\right)}\frac{x^{k}}{k!}+C\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k-1\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k-1\right)}\frac{x^{k}}{k!}+C\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}\left(a_{i}-1\right)^{-1}Q\left(a_{i}-1,k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}\left(b_{j}-1\right)^{-1}Q\left(b_{j}-1,k\right)}\frac{x^{k}}{k!}+C\\ & =\frac{\prod_{j=1}^{\dim\boldsymbol{b}}\left(b_{j}-1\right)}{\prod_{i=1}^{\dim\boldsymbol{a}}\left(a_{i}-1\right)}F\left(\boldsymbol{a}-\boldsymbol{1};\boldsymbol{b}-\boldsymbol{1};x\right)+C \end{align*}(4)
\begin{align*} \int x^{u}F\left(\boldsymbol{a};\boldsymbol{b};cx^{v}\right)dx & =\int x^{u}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{\left(cx^{v}\right)^{k}}{k!}dx\\ & =\int\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{c^{k}x^{u+vk}}{k!}dx\\ & =\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{c^{k}x^{u+vk+1}}{\left(u+vk+1\right)k!}+C\\ & =x^{u+1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{\left(cx^{v}\right)^{k}}{v\left(k+\frac{u+1}{v}\right)k!}+C\\ & =x^{u+1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{\frac{v}{u+1}Q\left(\frac{u+1}{v},k\right)\left(cx^{v}\right)^{k}}{vQ\left(\frac{u+1}{v}+1,k\right)k!}+C\\ & =\frac{x^{u+1}}{u+1}\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{\dim\boldsymbol{a}}Q\left(a_{i},k\right)}{\prod_{j=1}^{\dim\boldsymbol{b}}Q\left(b_{j},k\right)}\frac{Q\left(\frac{u+1}{v},k\right)\left(cx^{v}\right)^{k}}{Q\left(\frac{u+1}{v}+1,k\right)k!}+C\\ & =\frac{x^{u+1}}{u+1}F\left(\boldsymbol{a},\frac{u+1}{v};\boldsymbol{b},\frac{u+1}{v}+1;cx^{v}\right)+C \end{align*}
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簡単な関数を上昇階乗とべき乗を使って表す
\[
ak+b=b\frac{Q\left(\frac{b}{a}+1,k\right)}{Q\left(\frac{b}{a},k\right)}
\]
超幾何微分方程式(ガウスの微分方程式)の解
\[
x\left(1-x\right)y''\left(x\right)+\left(c-\left(a+b+1\right)x\right)y'\left(x\right)-aby\left(x\right)=0
\]
基本的な関数の一般化超幾何関数表示
\[
\left(1+x\right)^{a}=F\left(-a;;-x\right)
\]
合流型超幾何微分方程式の解
\[
xy''\left(x\right)+\left(b-x\right)y'\left(x\right)-ay\left(x\right)=0
\]