(1)

\begin{align*} \frac{1}{n!}\left[\frac{d^{n}}{dz^{n}}G\left(z\right)\right]_{z=0} & =\frac{1}{n!}\left[\frac{d^{n}}{dz^{n}}\sum_{k=0}^{\infty}a_{k}z^{k}\right]_{z=0}\\ & =a_{k}\frac{1}{n!}\sum_{k=0}^{\infty}\left[\frac{d^{n}}{dz^{n}}z^{k}\right]_{z=0}\\ & =a_{k}\frac{1}{n!}\sum_{k=0}^{\infty}k!\delta_{n,k}\\ & =a_{k} \end{align*} より題意は成り立つ。

(2)

\begin{align*} \left[\frac{d^{n}}{dz^{n}}G_{E}\left(z\right)\right]_{z=0} & =\left[\frac{d^{n}}{dz^{n}}\sum_{k=0}^{\infty}\frac{a_{k}}{k!}z^{k}\right]_{z=0}\\ & =\sum_{k=0}^{\infty}\frac{a_{k}}{k!}\left[\frac{d^{n}}{dz^{n}}z^{k}\right]_{z=0}\\ & =\sum_{k=0}^{\infty}\frac{a_{k}}{k!}k!\delta_{n,k}\\ & =a_{n} \end{align*} より題意は成り立つ。

(3)

\begin{align*} \left[n!\frac{d^{n}}{dz^{n}}e^{z}G_{P}\left(z\right)\right]_{z=0} & =\left[n!\frac{d^{n}}{dz^{n}}e^{z}\sum_{k=0}^{\infty}a_{k}e^{-z}\frac{z^{k}}{k!}\right]_{z=0}\\ & =n!\sum_{k=0}^{\infty}\frac{a_{k}}{k!}\left[\frac{d^{n}}{dz^{n}}z^{k}\right]_{z=0}\\ & =n!\sum_{k=0}^{\infty}\frac{a_{k}}{k!}\delta_{n,k}\\ & =\sum_{k=0}^{\infty}a_{k}\delta_{n,k}\\ & =a_{n} \end{align*} より題意は成り立つ。