(1)

第1種スターリング数の定義なので成り立つ。

(1)-2

\begin{align*} \sum_{k=0}^{\infty}S_{1}\left(n,k\right)x^{k} & =\sum_{j=0}^{\infty}P\left(x,j\right)\sum_{k=0}^{\infty}S_{1}\left(n,k\right)S_{2}\left(k,j\right)\\ & =\sum_{j=0}^{\infty}P\left(x,j\right)\delta_{n,j}\\ & =P\left(x,n\right) \end{align*}

(2)

\begin{align*} \sum_{n=0}^{\infty}S_{1}\left(n,k\right)\frac{x^{n}}{n!} & =\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}S_{1}\left(n,j\right)\frac{x^{n}}{n!}\delta_{j,k}\\ & =\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}S_{1}\left(n,j\right)\frac{x^{n}}{n!}\frac{P\left(j,k\right)}{k!}\delta_{j,k}\\ & =\left[\sum_{j=0}^{\infty}\sum_{n=0}^{\infty}S_{1}\left(n,j\right)\frac{x^{n}}{n!}\frac{P\left(j,k\right)}{k!}\frac{1}{P\left(j,k\right)}\frac{\partial^{k}}{\partial y^{k}}y^{j}\right]_{y\rightarrow0}\\ & =\frac{1}{k!}\left[\frac{\partial^{k}}{\partial y^{k}}\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\sum_{j=0}^{\infty}S_{1}\left(n,j\right)y^{j}\right]_{y\rightarrow0}\\ & =\frac{1}{k!}\left[\frac{\partial^{k}}{\partial y^{k}}\sum_{n=0}^{\infty}P\left(y,n\right)\frac{x^{n}}{n!}\right]_{y\rightarrow0}\\ & =\frac{1}{k!}\left[\frac{\partial^{k}}{\partial y^{k}}\left(1+x\right)^{y}\right]_{y\rightarrow0}\\ & =\frac{1}{k!}\left[\left(1+x\right)^{y}\log^{k}\left(1+x\right)\right]_{y\rightarrow0}\\ & =\frac{\log^{k}\left(1+x\right)}{k!} \end{align*}

(3)

\begin{align*} \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}S_{1}\left(n,k\right)\frac{x^{n}}{n!}y^{k} & =\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\sum_{k=0}^{\infty}S_{1}\left(n,k\right)y^{k}\\ & =\sum_{n=0}^{\infty}\frac{x^{n}}{n!}P\left(y,n\right)\\ & =\sum_{n=0}^{\infty}C\left(y,n\right)x^{n}\\ & =\left(1+x\right)^{y} \end{align*}

(4)

\begin{align*} \sum_{n=0}^{\infty}S_{2}\left(n,k\right)x^{n} & =\sum_{n=0}^{\infty}\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)j^{n}x^{n}\\ & =\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)\sum_{n=0}^{\infty}j^{n}x^{n}\\ & =\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)\frac{1}{1-jx}\\ & =\frac{\left(-1\right)^{k}}{k!}\sum_{j=0}^{k}\left(-1\right)^{j}C\left(k,j\right)\frac{\frac{1}{x}}{\frac{1}{x}-j}\\ & =\frac{\left(-1\right)^{k}}{xk!}\frac{\left(-1\right)^{k}}{\left(\frac{1}{x}-k\right)C\left(\frac{1}{x},k\right)}\\ & =\frac{\left(\frac{1}{x}-k\right)!}{x\left(\frac{1}{x}-k\right)\frac{1}{x}!}\\ & =\frac{1}{x}\frac{\Gamma\left(\frac{1}{x}-k\right)}{\Gamma\left(1+\frac{1}{x}\right)}\\ & =\frac{1}{xk!}B\left(\frac{1}{x}-k,k+1\right) \end{align*} 更に計算を進めると、
\begin{align*} \sum_{n=0}^{\infty}S_{2}\left(n,k\right)x^{n} & =\frac{1}{x}\frac{\Gamma\left(\frac{1}{x}-k\right)}{\Gamma\left(1+\frac{1}{x}\right)}\\ & =\frac{1}{x}\Gamma\left(\frac{1}{x}-k\right)\left(\Gamma\left(\frac{1}{x}-k\right)\prod_{j=1}^{k+1}\left(\frac{1}{x}+1-j\right)\right)^{-1}\\ & =\frac{1}{x}\left(\prod_{j=1}^{k+1}\left(\frac{1}{x}+1-j\right)\right)^{-1}\\ & =\frac{1}{x}\left(\prod_{j=0}^{k}\left(\frac{1}{x}-j\right)\right)^{-1}\\ & =\frac{1}{x}\left(\frac{1}{x^{k+1}}\prod_{j=0}^{k}\left(1-jx\right)\right)^{-1}\\ & =x^{k}\prod_{j=1}^{k}\left(1-jx\right)^{-1} \end{align*} となる。

(5)

\begin{align*} \sum_{n=0}^{\infty}S_{2}\left(n,k\right)\frac{x^{n}}{n!} & =\sum_{n=0}^{\infty}\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)j^{n}\frac{x^{n}}{n!}\\ & =\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)\sum_{n=0}^{\infty}\frac{\left(jx\right)^{n}}{n!}\\ & =\frac{1}{k!}\sum_{j=0}^{k}\left(-1\right)^{k-j}C\left(k,j\right)e^{jx}\\ & =\frac{1}{k!}\left(e^{x}-1\right)^{k} \end{align*}

(6)

\begin{align*} \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}S_{2}\left(n,k\right)\frac{x^{n}}{n!}y^{k} & =\sum_{k=0}^{\infty}\frac{\left(e^{x}-1\right)^{k}}{k!}y^{k}\\ & =\sum_{k=0}^{\infty}\frac{\left(y\left(e^{x}-1\right)\right)^{k}}{k!}\\ & =\exp\left(y\left(e^{x}-1\right)\right) \end{align*}

(7)

第2種スターリング数の定義なので成り立つ。

(7)-2

\begin{align*} \sum_{k=0}^{\infty}S_{2}\left(n,k\right)P\left(x,k\right) & =\sum_{k=0}^{\infty}S_{2}\left(n,k\right)\sum_{j=0}^{\infty}S_{1}\left(k,j\right)x^{j}\\ & =\sum_{j=0}^{\infty}x^{j}\sum_{k=0}^{\infty}S_{2}\left(n,k\right)S_{1}\left(k,j\right)\\ & =\sum_{j=0}^{\infty}x^{j}\delta_{n,j}\\ & =x^{n} \end{align*}