(1)

\begin{align*} a_{n} & =\sum_{k=0}^{n}\delta_{nk}a_{k}\\ & =\sum_{k=0}^{n}\sum_{j=0}^{n}\left(-1\right)^{j+n}C(n,j)C(j,k)a_{k}\\ & =\sum_{j=0}^{n}\left(-1\right)^{j+n}C(n,j)b_{j}\\ & =\sum_{j=0}^{n}\left(-1\right)^{n-j}C(n,j)b_{j}\qquad\tag{*}\\ & =\sum_{j=0}^{n}\left(-1\right)^{j}C(n,j)b_{n-j} \end{align*}

(1)-2

母関数を定義する。

\begin{align*} A(x) & =\sum_{k=0}^{\infty}\frac{a_{k}x^{k}}{k!}\\ B(x) & =\sum_{k=0}^{\infty}\frac{b_{k}x^{k}}{k!} \end{align*}

これを使うと、

\begin{align*} B(x) & =\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\sum_{j=0}^{k}C(k,j)a_{j}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}\frac{a_{j}x^{j}}{j!}\frac{x^{k-j}}{(k-j)!}\\ & =\sum_{m=0}^{\infty}\sum_{j=0}^{\infty}\frac{a_{j}x^{j}}{j!}\frac{x^{m}}{m!}\\ & =A(x)e^{x} \end{align*}

これより、

\begin{align*} A(x) & =B(x)e^{-x}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\frac{b_{k}x^{k}}{k!}\frac{(-x)^{j}}{j!}\\ & =\sum_{m=0}^{\infty}\sum_{k=0}^{m}\frac{b_{k}x^{k}}{k!}\frac{(-x)^{m-k}}{(m-k)!}\\ & =\sum_{m=0}^{\infty}\left(\sum_{k=0}^{m}C(m,k)(-1)^{m-k}b_{k}\right)\frac{x^{m}}{m!} \end{align*}

となるので、

\begin{align*} a_{m} & =\sum_{k=0}^{m}C(m,k)(-1)^{m-k}b_{k}\\ & =\sum_{k=0}^{m}C(m,k)(-1)^{k}b_{m-k} \end{align*}

(2)

\begin{align*} a_{n} & =\sum_{k=0}^{n}\delta_{nk}a_{k}\\ & =\sum_{k=0}^{n}(-1)^{n+k}\delta_{nk}a_{k}\\ & =\sum_{k=0}^{n}\sum_{j=0}^{n}\left(-1\right)^{n+k}\left(-1\right)^{n+j}C\left(n,j\right)C\left(j,k\right)a_{k}\\ & =\sum_{j=0}^{n}\left(-1\right)^{j}C\left(n,j\right)b_{j} \end{align*}