ベル数の指数型母関数

\begin{align*} \sum_{k=0}^{\infty}B\left(k\right)\frac{x^{k}}{k!} & =\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}S_{2}\left(k,j\right)\frac{x^{k}}{k!}\\ & =\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\sum_{j=0}^{\infty}S_{2}\left(k,j\right)\\ & =\left[e^{-t}\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\sum_{j=0}^{\infty}S_{2}\left(k,j\right)t^{j}\left(\frac{d}{dt}\right)^{j}e^{t}\right]_{t=1}\\ & =\left[e^{-t}\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\left(t\frac{d}{dt}\right)^{k}e^{t}\right]_{t=1}\cmt{\because\left(x\frac{d}{dx}\right)^{n}=\sum_{k=0}^{\infty}S_{2}\left(n,k\right)x^{k}\frac{d^{k}}{dx^{k}}}\\ & =\left[e^{-t}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{x^{k}}{k!}\left(t\frac{d}{dt}\right)^{k}\frac{t^{j}}{j!}\right]_{t=1}\\ & =\left[e^{-t}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{x^{k}}{k!}\frac{j^{k}t^{j}}{j!}\right]_{t=1}\\ & =\left[e^{-t}\sum_{j=0}^{\infty}\frac{t^{j}}{j!}\sum_{k=0}^{\infty}\frac{\left(xj\right)^{k}}{k!}\right]_{t=1}\\ & =\left[e^{-t}\sum_{j=0}^{\infty}\frac{t^{j}}{j!}e^{xj}\right]_{t=1}\\ & =\left[e^{-t}\sum_{j=0}^{\infty}\frac{1}{j!}\left(te^{x}\right)^{j}\right]_{t=1}\\ & =\left[e^{-t}e^{te^{x}}\right]_{t=1}\\ & =\left[e^{te^{x}-t}\right]_{t=1}\\ & =e^{e^{x}-1} \end{align*}