オイラー多項式の指数型母関数

\begin{align*} \sum_{k=0}^{\infty}\frac{E_{k}\left(x\right)}{k!}t^{k} & =\sum_{k=0}^{\infty}\left(\sum_{j=0}^{k}C\left(k,j\right)\frac{E_{j}}{2^{j}}\left(x-\frac{1}{2}\right)^{k-j}\right)\frac{t^{k}}{k!}\\ & =\sum_{j=0}^{\infty}\sum_{k=j}^{\infty}C\left(k,j\right)\frac{E_{j}}{2^{j}}\left(x-\frac{1}{2}\right)^{k-j}\frac{t^{k}}{k!}\\ & =\sum_{j=0}^{\infty}\sum_{k=j}^{\infty}\frac{E_{j}}{2^{j}j!}\left(x-\frac{1}{2}\right)^{k-j}\frac{t^{j}t^{k-j}}{\left(k-j\right)!}\\ & =\sum_{j=0}^{\infty}\frac{E_{j}}{2^{j}j!}t^{j}\sum_{k=0}^{\infty}\left(x-\frac{1}{2}\right)^{k}\frac{t^{k}}{k!}\\ & =\sum_{j=0}^{\infty}\frac{E_{j}}{j!}\left(\frac{t}{2}\right)^{j}\sum_{k=0}^{\infty}\frac{\left(\left(x-\frac{1}{2}\right)t\right)^{k}}{k!}\\ & =\cosh^{-1}\left(\frac{t}{2}\right)e^{\left(x-\frac{1}{2}\right)t}\\ & =\frac{2e^{\left(x-\frac{1}{2}\right)t}}{e^{\frac{t}{2}}+e^{-\frac{t}{2}}}\\ & =\frac{2e^{xt}}{e^{t}+1} \end{align*}