ガウスの乗法公式

ガウスの乗法公式

\(n\in\mathbb{N}\)とする。

\[ \Gamma(nz)=\frac{n^{nz-\frac{1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right) \]

(0)

\begin{align*} \Gamma(nz) & =\Gamma(nz)\prod_{m=0}^{r-1}\prod_{j=0}^{n-1}\left(\frac{1}{n}\frac{n\left(z+m\right)+j}{z+m+\frac{j}{n}}\right)\qquad,\qquad\forall r\in\mathbb{N}_{0}\\ & =\Gamma(nz)\frac{1}{n^{nr}}\prod_{m=0}^{r-1}\left(\frac{\Gamma(n\left(z+m\right)+n)}{\Gamma(n\left(z+m\right))}\prod_{j=0}^{n-1}\left(\frac{1}{z+\frac{j}{n}+m}\right)\right)\\ & =\Gamma(nz)\frac{1}{n^{nr}}\prod_{m=0}^{r-1}\left(\frac{\Gamma(n\left(z+m+1\right))}{\Gamma(n\left(z+m\right))}\prod_{j=0}^{n-1}\left(\frac{1}{z+\frac{j}{n}+m}\right)\right)\\ & =\Gamma(nz)\frac{1}{n^{nr}}\frac{\Gamma(n\left(z+r\right))}{\Gamma(nz)}\prod_{j=0}^{n-1}\prod_{m=0}^{r-1}\frac{1}{z+\frac{j}{n}+m}\\ & =\frac{\Gamma(n\left(z+r\right))}{n^{nr}}\prod_{j=0}^{n-1}\frac{\Gamma\left(z+\frac{j}{n}\right)}{\Gamma\left(z+\frac{j}{n}+r\right)}\\ & =\frac{\Gamma(n\left(z+r\right))}{n^{nr}}\left(\prod_{j=0}^{n-1}\frac{1}{\Gamma\left(z+\frac{j}{n}+r\right)}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\lim_{r\rightarrow\infty}\frac{\sqrt{\frac{2\pi}{n\left(z+r\right)}}\left(\frac{n\left(z+r\right)}{e}\right)^{n\left(z+r\right)}}{n^{nr}}\left(\prod_{j=0}^{n-1}\sqrt{\frac{z+\frac{j}{n}+r}{2\pi}}\left(\frac{e}{z+\frac{j}{n}+r}\right)^{z+\frac{j}{n}+r}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\qquad,\qquad\text{スターリングの公式}\\ & =\lim_{r\rightarrow\infty}\frac{\sqrt{2\pi}e^{-n\left(z+r\right)}\left(n\left(z+r\right)\right)^{n\left(z+r\right)-\frac{1}{2}}}{n^{nr}}\left(\prod_{j=0}^{n-1}\left(2\pi\right)^{-\frac{1}{2}}\left(z+\frac{j}{n}+r\right)^{-z-\frac{j}{n}-r+\frac{1}{2}}e^{z+\frac{j}{n}+r}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\lim_{r\rightarrow\infty}\frac{n^{nz-\frac{1}{2}}\left(z+r\right)^{n\left(z+r\right)-\frac{1}{2}}e^{\frac{n-1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\left(\prod_{j=0}^{n-1}\left(z+\frac{j}{n}+r\right)^{-z-\frac{j}{n}-r+\frac{1}{2}}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\lim_{r\rightarrow\infty}\frac{n^{nz-\frac{1}{2}}\left(z+r\right)^{n\left(z+r\right)-\frac{1}{2}}e^{\frac{n-1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\left(\prod_{j=0}^{n-1}\left(z+r\right)^{-z-\frac{j}{n}-r+\frac{1}{2}}\left(1+\frac{j}{n\left(z+r\right)}\right)^{-(z+r)}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\lim_{r\rightarrow\infty}\frac{n^{nz-\frac{1}{2}}\left(z+r\right)^{n\left(z+r\right)-\frac{1}{2}}e^{\frac{n-1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\left(\prod_{j=0}^{n-1}\left(z+r\right)^{-z-\frac{j}{n}-r+\frac{1}{2}}e^{-\frac{j}{n}}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\lim_{r\rightarrow\infty}\frac{n^{nz-\frac{1}{2}}\left(z+r\right)^{n\left(z+r\right)-\frac{1}{2}}e^{\frac{n-1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\left(\left(z+r\right)^{-zn-\frac{n-1}{2}-rn+\frac{n}{2}}e^{-\frac{n-1}{2}}\right)\left(\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right)\right)\\ & =\frac{n^{nz-\frac{1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\prod_{j=0}^{n-1}\Gamma\left(z+\frac{j}{n}\right) \end{align*}

(0)-2

\(z\in\mathbb{N}\)のときの証明

\(m=z\)とおく。

\begin{align*} \Gamma\left(nm\right) & =\frac{1}{nm}\prod_{j=0}^{m-1}\prod_{k=0}^{n-1}\left(nj+k+1\right)\\ & =\frac{n^{nm}}{nm}\prod_{j=0}^{m-1}\prod_{k=0}^{n-1}\left(j+\frac{k+1}{n}\right)\\ & =\frac{n^{nm}}{nm}\left\{ \prod_{k=0}^{n-1}\prod_{j=0}^{m-1}\left(j+\frac{k+1}{n}\right)\right\} \left\{ \prod_{k=0}^{n-1}\Gamma\left(\frac{k+1}{n}\right)\Gamma^{-1}\left(\frac{k+1}{n}\right)\right\} \\ & =\frac{n^{nm}}{nm}\left\{ \prod_{k=0}^{n-1}\Gamma\left(\frac{k+1}{n}\right)\prod_{j=0}^{m-1}\left(j+\frac{k+1}{n}\right)\right\} \left\{ \Gamma\left(1\right)\sqrt{\prod_{k=0}^{n-2}\Gamma^{-1}\left(\frac{k+1}{n}\right)\Gamma^{-1}\left(\frac{n-2-k+1}{n}\right)}\right\} \\ & =\frac{n^{nm}}{nm}\left\{ \prod_{k=0}^{n-1}\Gamma\left(m+\frac{k+1}{n}\right)\right\} \left\{ \sqrt{\prod_{k=0}^{n-2}\frac{\sin\left(\frac{k+1}{n}\pi\right)}{\pi}}\right\} \\ & =\frac{n^{nm}}{nm}\left\{ m\prod_{k=0}^{n-1}\Gamma\left(m+\frac{k}{n}\right)\right\} \left\{ \sqrt{\frac{n}{\pi^{n-1}2^{n-1}}}\right\} \\ & =\frac{n^{nm-\frac{1}{2}}}{\left(2\pi\right)^{\frac{n-1}{2}}}\prod_{k=0}^{n-1}\Gamma\left(m+\frac{k}{n}\right) \end{align*}

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ガウスの乗法公式

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