(0)

\[ \begin{cases} \psi^{\left(n\right)}\left(z+1\right)=\psi^{\left(n\right)}\left(z\right)+\frac{\left(-1\right)^{n}n!}{z^{n+1}} & n\in\mathbb{N}_{0}\\ \psi\left(1\right)=-\gamma\\ \psi^{\left(n\right)}\left(1\right)=\left(-1\right)^{n+1}n!\zeta\left(n+1\right) & n\in\mathbb{N}\\ \psi\left(z\right)=-\gamma+H_{z-1} & z\in\mathbb{Z}\setminus\mathbb{N}_{0}^{-}\\ \psi^{\left(n\right)}\left(z\right)=\left(-1\right)^{n}n!\left\{ -\zeta\left(n+1\right)+H_{z-1,n+1}\right\} & n\in\mathbb{N},z\in\mathbb{Z}\setminus\mathbb{N}_{0}^{-}\\ \lim_{z\rightarrow0}\left(\psi^{\left(n\right)}\left(z-m\right)-\psi^{\left(n\right)}\left(z\right)\right)=n!H_{m,n+1} & m,n\in\mathbb{N}_{0} \end{cases} \] \[ \begin{cases} P\left(x,y\right)=Q\left(x-y+1,y\right)\\ Q\left(x,y\right)=P\left(x+y-1,y\right) \end{cases} \] を使う。

(1)

\begin{align*} \left[\frac{d^{0}}{dx^{0}}Q\left(x,y\right)\right]_{x\rightarrow0} & =\left[\frac{\Gamma\left(x+y\right)}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\begin{cases} 0 & y\notin\mathbb{N}_{0}^{-}\\ \left[\frac{\left(-1\right)^{y}}{\left(-y\right)!}\frac{\Gamma\left(x\right)}{\Gamma\left(x\right)}\right]_{x\rightarrow0} & y\in\mathbb{N}_{0}^{-} \end{cases}\\ & =\begin{cases} 0 & y\notin\mathbb{N}_{0}^{-}\\ \frac{\left(-1\right)^{y}}{\left(-y\right)!} & y\in\mathbb{N}_{0}^{-} \end{cases} \end{align*}

(2)

\begin{align*} \frac{d}{dx}Q\left(x,y\right) & =\frac{\Gamma\left(x+y\right)\psi\left(x+y\right)}{\Gamma\left(x\right)}-\frac{\Gamma\left(x+y\right)\psi\left(x\right)}{\Gamma\left(x\right)}\\ & =\Gamma\left(x+y\right)\frac{\psi\left(x+y\right)-\psi\left(x\right)}{\Gamma\left(x\right)}\\ & =Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \end{align*}

(3)

\(y\notin\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d}{dx}Q\left(x,y\right)\right]_{x\rightarrow0} & =\left[Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+y\right)\frac{\psi\left(x+y\right)-\psi\left(x\right)}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{\psi\left(x+y\right)-\psi\left(x+1\right)+\frac{1}{x}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{x\left\{ \psi\left(x+y\right)-\psi\left(x+1\right)\right\} +1}{\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[x\left\{ \psi\left(y\right)-\psi\left(1\right)\right\} +1\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[x\left\{ \psi\left(y\right)+\gamma\right\} +1\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right) \end{align*}

\(y=-n\in\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d}{dx}Q\left(x,-n\right)\right]_{x\rightarrow0} & =\left[Q\left(x,-n\right)\left\{ \psi\left(x-n\right)-\psi\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[P^{-1}\left(x-1,n\right)\left\{ \psi\left(x-n\right)-\psi\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}\left[\psi\left(x-n\right)-\psi\left(x\right)\right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}H_{n} \end{align*}

-

これより、
\[ \left[\frac{d}{dx}Q\left(x,y\right)\right]_{x\rightarrow0}=\begin{cases} \Gamma\left(y\right) & y\notin\mathbb{N}_{0}^{-}\\ \frac{\left(-1\right)^{y}}{\left(-y\right)!}H_{-y} & y\in\mathbb{N}_{0}^{-} \end{cases} \] となる。

(4)

\begin{align*} \frac{d^{2}Q\left(x,y\right)}{dx^{2}} & =\frac{d}{dx}\left(Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \right)\\ & =Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} +Q\left(x,y\right)\left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \\ & =Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \end{align*}

(5)

\(y\notin\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d^{2}Q\left(x,y\right)}{dx^{2}}\right]_{x\rightarrow0} & =\left[Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+y\right)\frac{\left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{\left(\psi\left(x+y\right)-\psi\left(x+1\right)+\frac{1}{x}\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)-\frac{1}{x^{2}}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{\left(x\psi\left(x+y\right)-x\psi\left(x+1\right)+1\right)^{2}+x^{2}\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)-1}{x\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{x^{2}\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{2}+2x\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)+1+x^{2}\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)-1}{x\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{x\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{2}+2\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)+x\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)}{\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =2\Gamma\left(y\right)\left(\psi\left(y\right)-\psi\left(1\right)\right)\\ & =2\Gamma\left(y\right)\left(\psi\left(y\right)+\gamma\right)\\ & =2\Gamma\left(y\right)H_{y-1} \end{align*}

\(y=-n\in\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d^{2}Q\left(x,-n\right)}{dx^{2}}\right]_{x\rightarrow0} & =\left[Q\left(x,-n\right)\left\{ \left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[P^{-1}\left(x-1,n\right)\left\{ \left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}\left[\left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}\left(H_{n,1}^{2}+H_{n,2}\right) \end{align*}

-

これより、
\[ \left[\frac{d^{2}Q\left(x,y\right)}{dx^{2}}\right]_{x\rightarrow0}=\begin{cases} 2\Gamma\left(y\right)H_{y-1} & y\notin\mathbb{N}_{0}^{-}\\ \frac{\left(-1\right)^{-y}}{\left(-y\right)!}\left(H_{-y,1}^{2}+H_{-y,2}\right) & y\in\mathbb{N}_{0}^{-} \end{cases} \] となる。

(6)

\begin{align*} \frac{d^{3}Q\left(x,y\right)}{dx^{3}} & =\frac{d}{dx}\left(Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right)\\ & =Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +Q\left(x,y\right)\left\{ 2\left(\psi\left(x+y\right)-\psi\left(x\right)\right)\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right)+\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \\ & =Q\left(x,y\right)\left\{ \psi\left(x+y\right)-\psi\left(x\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +2\left(\psi\left(x+y\right)-\psi\left(x\right)\right)\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right)+\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \\ & =Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \end{align*}

(7)

\(y\notin\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d^{3}Q\left(x,y\right)}{dx^{3}}\right]_{x\rightarrow0} & =\left[Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+y\right)\frac{\left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{\left(\psi\left(x+y\right)-\psi\left(x+1\right)+\frac{1}{x}\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x+1\right)+\frac{1}{x}\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)-\frac{1}{x^{2}}\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x+1\right)+\frac{2}{x^{3}}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{\left(x\psi\left(x+y\right)-x\psi\left(x+1\right)+1\right)^{3}+3\left\{ x\psi\left(x+y\right)-x\psi\left(x+1\right)+1\right\} \left\{ x^{2}\psi^{\left(1\right)}\left(x+y\right)-x^{2}\psi^{\left(1\right)}\left(x+1\right)-1\right\} +x^{3}\left(\psi^{\left(2\right)}\left(x+y\right)-x\psi^{\left(2\right)}\left(x+1\right)\right)+2}{x^{2}\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{x^{3}\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{3}+3x^{2}\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{2}+3x\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)+1+3\left\{ x^{3}\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)+x^{2}\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)-x\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)-1\right\} +x^{3}\left(\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x+1\right)\right)+2}{x^{2}\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(y\right)\left[\frac{x^{3}\left\{ \left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{3}+3\left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)+\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x+1\right)\right\} +3x^{2}\left\{ \left(\psi\left(x+y\right)-\psi\left(x+1\right)\right)^{2}+\left(\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x+1\right)\right)\right\} }{x^{2}\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =3\Gamma\left(y\right)\left\{ \left(\psi\left(y\right)-\psi\left(1\right)\right)^{2}+\left(\psi^{\left(1\right)}\left(y\right)-\psi^{\left(1\right)}\left(1\right)\right)\right\} \\ & =3\Gamma\left(y\right)\left\{ \left(\psi\left(y\right)+\gamma\right)^{2}+\left(\psi^{\left(1\right)}\left(y\right)-\zeta\left(2\right)\right)\right\} \\ & =3\Gamma\left(y\right)\left(H_{y-1}^{2}-H_{y-1,2}\right) \end{align*}

\(y=-n\in\mathbb{N}_{0}^{-}\)のとき、

\begin{align*} \left[\frac{d^{3}Q\left(x,-n\right)}{dx^{3}}\right]_{x\rightarrow0} & =\left[Q\left(x,-n\right)\left\{ \left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x-n\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x-n\right)-\psi^{\left(2\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\left[P^{-1}\left(x-1,n\right)\left\{ \left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x-n\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x-n\right)-\psi^{\left(2\right)}\left(x\right)\right\} \right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}\left[\left(\psi\left(x-n\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x-n\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x-n\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x-n\right)-\psi^{\left(2\right)}\left(x\right)\right]_{x\rightarrow0}\\ & =\frac{\left(-1\right)^{n}}{n!}\left(H_{n,1}^{3}+3H_{n,1}H_{n,2}+2H_{n,3}\right) \end{align*}

-

これより、
\[ \left[\frac{d^{3}Q\left(x,y\right)}{dx^{3}}\right]_{x\rightarrow0}=\begin{cases} 3\Gamma\left(y\right)\left(H_{y-1}^{2}-H_{y-1,2}\right) & y\notin\mathbb{N}_{0}^{-}\\ \frac{\left(-1\right)^{-y}}{\left(-y\right)!}\left(H_{-y,1}^{3}+3H_{-y,1}H_{-y,2}+2H_{-y,3}\right) & y\in\mathbb{N}_{0}^{-} \end{cases} \] となる。

(8)

\begin{align*} \left[\frac{d^{0}}{dx^{0}}P\left(x,y\right)\right]_{x\rightarrow0} & =\left[\frac{\Gamma\left(x+1\right)}{\Gamma\left(x-y+1\right)}\right]_{x\rightarrow0}\\ & =\begin{cases} \frac{1}{\left(-y\right)!} & y\notin\mathbb{N}\\ 0 & y\in\mathbb{N} \end{cases} \end{align*}

(9)

(2)より、
\begin{align*} \frac{d}{dx}P\left(x,y\right) & =\frac{d}{dx}Q\left(x-y+1,y\right)\\ & =\left[Q\left(x,y\right)\psi\left(x+y\right)-\psi\left(x\right)\right]_{x\rightarrow x-y+1}\\ & =\left[P\left(x+y-1,y\right)\psi\left(x+y\right)-\psi\left(x\right)\right]_{x\rightarrow x-y+1}\\ & =P\left(x,y\right)\left\{ \psi\left(x+1\right)-\psi\left(x-y+1\right)\right\} \end{align*} となるので与式は成り立つ。

(10)

\(y\notin\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d}{dx}P\left(x,y\right)\right]_{x\rightarrow0} & =\left[P\left(x,y\right)\left\{ \psi\left(x+1\right)-\psi\left(x-y+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\psi\left(x+1\right)-\psi\left(x-y+1\right)}{\Gamma\left(x-y+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\frac{\psi\left(1\right)-\psi\left(1-y\right)}{\Gamma\left(1-y\right)}\\ & =\frac{\gamma-\psi\left(1-y\right)}{\Gamma\left(1-y\right)} \end{align*}

\(y=n\in\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d}{dx}P\left(x,n\right)\right]_{x\rightarrow0} & =\left[P\left(x,n\right)\left\{ \psi\left(x+1\right)-\psi\left(x-n+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\psi\left(x+1\right)-\psi\left(x-n+1\right)}{\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\left[\left(-1\right)^{n-1}\left(n-1\right)!\frac{\psi\left(x+1\right)-\left(\psi\left(x\right)+H_{n-1}\right)}{\left(-1\right)^{n-1}\left(n-1\right)!\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\frac{1}{x}-H_{n-1}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{1-xH_{n-1}}{x\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{1-xH_{n-1}}{\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)! \end{align*}

-

これより、
\[ \left[\frac{d}{dx}P\left(x,n\right)\right]_{x\rightarrow0}=\begin{cases} \frac{\gamma-\psi\left(1-y\right)}{\Gamma\left(1-y\right)} & y\notin\mathbb{N}\\ \left(-1\right)^{y-1}\left(y-1\right)! & y\in\mathbb{N} \end{cases} \] となる。

(11)

(4)より、
\begin{align*} \frac{d^{2}}{dx^{2}}P\left(x,y\right) & =\frac{d^{2}}{dx^{2}}Q\left(x-y+1,y\right)\\ & =\left[Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right]_{x\rightarrow x-y+1}\\ & =\left[P\left(x+y-1,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{2}+\psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} \right]_{x\rightarrow x-y+1}\\ & =P\left(x,y\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)\right\} \end{align*} となるので与式は成り立つ。

(12)

\(y\notin\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d^{2}}{dx^{2}}P\left(x,y\right)\right]_{x\rightarrow0} & =\left[P\left(x,y\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)}{\Gamma\left(x-y+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\frac{\left(\psi\left(1\right)-\psi\left(1-y\right)\right)^{2}+\psi^{\left(1\right)}\left(1\right)-\psi^{\left(1\right)}\left(1-y\right)}{\Gamma\left(1-y\right)}\\ & =\frac{\left(-\gamma-\psi\left(1-y\right)\right)^{2}+\zeta\left(2\right)-\psi^{\left(1\right)}\left(1-y\right)}{\Gamma\left(1-y\right)}\\ & =\frac{\left(\gamma+\psi\left(1-y\right)\right)^{2}+\frac{\pi^{2}}{6}-\psi^{\left(1\right)}\left(1-y\right)}{\Gamma\left(1-y\right)} \end{align*}

\(y=n\in\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d^{2}}{dx^{2}}P\left(x,n\right)\right]_{x\rightarrow0} & =\left[P\left(x,n\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-n+1\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-n+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\left(\psi\left(x+1\right)-\psi\left(x-n+1\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-n+1\right)}{\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\left[\left(-1\right)^{n-1}\left(n-1\right)!\frac{\left(\psi\left(x+1\right)-\left(\psi\left(x\right)+H_{n-1}\right)\right)^{2}+\psi^{\left(1\right)}\left(x+1\right)-\left(\psi^{\left(1\right)}\left(x\right)+H_{n-1,2}\right)}{\left(-1\right)^{n-1}\left(n-1\right)!\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(\frac{1}{x}-H_{n-1}\right)^{2}-\frac{1}{x^{2}}-H_{n-1,2}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(1-xH_{n-1}\right)^{2}-1-x^{2}H_{n-1,2}}{x^{2}\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{1-2xH_{n-1}+x^{2}H_{n-1}^{2}-1-x^{2}H_{n-1,2}}{x\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{-2xH_{n-1}+x^{2}\left(H_{n-1}^{2}-H_{n-1,2}\right)}{x}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[-2H_{n-1}+x\left(H_{n-1}^{2}-H_{n-1,2}\right)\right]_{x\rightarrow0}\\ & =-2H_{n-1}\left(-1\right)^{n-1}\left(n-1\right)! \end{align*}

-

これより、
\[ \left[\frac{d^{2}}{dx^{2}}P\left(x,n\right)\right]_{x\rightarrow0}=\begin{cases} \frac{\left(\gamma+\psi\left(1-y\right)\right)^{2}+\frac{\pi^{2}}{6}-\psi^{\left(1\right)}\left(1-y\right)}{\Gamma\left(1-y\right)} & y\notin\mathbb{N}\\ -2H_{n-1}\left(-1\right)^{n-1}\left(n-1\right)! & y\in\mathbb{N} \end{cases} \] となる。

(13)

(6)より、
\begin{align*} \frac{d^{3}}{dx^{3}}P\left(x,y\right) & =\frac{d^{3}}{dx^{3}}Q\left(x-y+1,y\right)\\ & =\left[Q\left(x,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \right]_{x\rightarrow x-y+1}\\ & =\left[P\left(x+y-1,y\right)\left\{ \left(\psi\left(x+y\right)-\psi\left(x\right)\right)^{3}+3\left\{ \psi\left(x+y\right)-\psi\left(x\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+y\right)-\psi^{\left(1\right)}\left(x\right)\right\} +\psi^{\left(2\right)}\left(x+y\right)-\psi^{\left(2\right)}\left(x\right)\right\} \right]_{x\rightarrow x-y+1}\\ & =P\left(x,y\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\psi\left(x-y+1\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\psi^{\left(2\right)}\left(x-y+1\right)\right\} \end{align*} となるので与式は成り立つ。

(14)

\(y\notin\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d^{3}}{dx^{3}}P\left(x,y\right)\right]_{x\rightarrow0} & =\left[P\left(x,y\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\psi\left(x-y+1\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\psi^{\left(2\right)}\left(x-y+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\left(\psi\left(x+1\right)-\psi\left(x-y+1\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\psi\left(x-y+1\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-y+1\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\psi^{\left(2\right)}\left(x-y+1\right)}{\Gamma\left(x-y+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\frac{\left(\psi\left(1\right)-\psi\left(1-y\right)\right)^{3}+3\left\{ \psi\left(1\right)-\psi\left(1-y\right)\right\} \left\{ \psi^{\left(1\right)}\left(1\right)-\psi^{\left(1\right)}\left(1-y\right)\right\} +\psi^{\left(2\right)}\left(1\right)-\psi^{\left(2\right)}\left(1-y\right)}{\Gamma\left(1-y\right)}\\ & =\frac{\left(-\gamma-\psi\left(1-y\right)\right)^{3}+3\left\{ -\gamma-\psi\left(1-y\right)\right\} \left\{ \zeta\left(2\right)-\psi^{\left(1\right)}\left(1-y\right)\right\} -2\zeta\left(3\right)-\psi^{\left(2\right)}\left(1-y\right)}{\Gamma\left(1-y\right)}\\ & =\frac{\left(\gamma+\psi\left(1-y\right)\right)^{3}-3\left\{ \gamma+\psi\left(1-y\right)\right\} \left\{ \frac{\pi^{2}}{6}-\psi^{\left(1\right)}\left(1-y\right)\right\} -2\zeta\left(3\right)-\psi^{\left(2\right)}\left(1-y\right)}{\Gamma\left(1-y\right)} \end{align*}

\(y=n\in\mathbb{N}\)のとき、

\begin{align*} \left[\frac{d^{3}}{dx^{3}}P\left(x,n\right)\right]_{x\rightarrow0} & =\left[P\left(x,n\right)\left\{ \left(\psi\left(x+1\right)-\psi\left(x-n+1\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\psi\left(x-n+1\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-n+1\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\psi^{\left(2\right)}\left(x-n+1\right)\right\} \right]_{x\rightarrow0}\\ & =\left[\Gamma\left(x+1\right)\frac{\left(\psi\left(x+1\right)-\psi\left(x-n+1\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\psi\left(x-n+1\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\psi^{\left(1\right)}\left(x-n+1\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\psi^{\left(2\right)}\left(x-n+1\right)}{\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\Gamma\left(1\right)\left[\left(-1\right)^{n-1}\left(n-1\right)!\frac{\left(\psi\left(x+1\right)-\left(\psi\left(x\right)+H_{n-1}\right)\right)^{3}+3\left\{ \psi\left(x+1\right)-\left(\psi\left(x\right)+H_{n-1}\right)\right\} \left\{ \psi^{\left(1\right)}\left(x+1\right)-\left(\psi^{\left(1\right)}\left(x\right)+H_{n-1,2}\right)\right\} +\psi^{\left(2\right)}\left(x+1\right)-\left(\psi^{\left(2\right)}\left(x\right)+2H_{n-1,3}\right)}{\left(-1\right)^{n-1}\left(n-1\right)!\Gamma\left(x-n+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(\frac{1}{x}-H_{n-1}\right)^{3}+3\left(\frac{1}{x}-H_{n-1}\right)\left(-\frac{1}{x^{2}}-H_{n-1,2}\right)+\frac{2}{x^{3}}-2H_{n-1,3}}{\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(1-xH_{n-1}\right)^{3}-3\left(1-xH_{n-1}\right)\left(1+x^{2}H_{n-1,2}\right)+2-2x^{3}H_{n-1,3}}{x^{3}\Gamma\left(x\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(1-xH_{n-1}\right)^{3}-3\left(1-xH_{n-1}\right)\left(1+x^{2}H_{n-1,2}\right)+2-2x^{3}H_{n-1,3}}{x^{2}\Gamma\left(x+1\right)}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{\left(1-xH_{n-1}\right)^{3}-3\left(1-xH_{n-1}\right)\left(1+x^{2}H_{n-1,2}\right)+2-2x^{3}H_{n-1,3}}{x^{2}}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{-3H_{n-1}\left(1-xH_{n-1}\right)^{2}+3H_{n-1}\left(1+x^{2}H_{n-1,2}\right)-6xH_{n-1,2}\left(1-xH_{n-1}\right)-6x^{2}H_{n-1,3}}{2x}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\left[\frac{6H_{n-1}^{2}\left(1-xH_{n-1}\right)+6xH_{n-1}H_{n-1,2}-6H_{n-1,2}\left(1-xH_{n-1}\right)+6xH_{n-1,2}H_{n-1}-12xH_{n-1,3}}{2}\right]_{x\rightarrow0}\\ & =\left(-1\right)^{n-1}\left(n-1\right)!\frac{6H_{n-1}^{2}-6H_{n-1,2}}{2}\\ & =\left(-1\right)^{n-1}3\left(n-1\right)!\left(H_{n-1}^{2}-H_{n-1,2}\right) \end{align*}

-

これより、
\[ \left[\frac{d^{3}}{dx^{3}}P\left(x,n\right)\right]_{x\rightarrow0}=\begin{cases} \frac{\left(\gamma+\psi\left(1-y\right)\right)^{3}-3\left\{ \gamma+\psi\left(1-y\right)\right\} \left\{ \frac{\pi^{2}}{6}-\psi^{\left(1\right)}\left(1-y\right)\right\} -2\zeta\left(3\right)-\psi^{\left(2\right)}\left(1-y\right)}{\Gamma\left(1-y\right)} & y\notin\mathbb{N}\\ \left(-1\right)^{n-1}3\left(n-1\right)!\left(H_{n-1}^{2}-H_{n-1,2}\right) & y\in\mathbb{N} \end{cases} \] となる。