(0)

\begin{align*} \delta_{mn} & =\left[\frac{1}{m!}\frac{dx^{m}}{dx^{n}}\right]_{x=0}\\ & =\left[\frac{1}{m!}\frac{d}{dx^{n}}(-1)^{m}(1-x)^{m}\right]_{x=1}\\ & =\left[\frac{1}{m!}(-1)^{m}\frac{d}{dx^{n}}\sum_{k=0}^{m}C(m,k)(-x)^{k}\right]_{x=1}\\ & =\left[\frac{1}{m!}(-1)^{m}\sum_{k=0}^{m}C(m,k)(-1)^{k}P(k,n)x^{k-n}\right]_{x=1}\\ & =\sum_{k=0}^{m}\frac{(-1)^{k+m}}{(m-k)!(k-n)!}\qquad\tag{*}\\ & =\frac{n!}{m!}\sum_{k=0}^{m}(-1)^{k+m}\frac{m!}{(m-k)!k!}\frac{k!}{(k-n)!n!}\\ & =\frac{n!}{m!}\sum_{k=0}^{m}(-1)^{k+m}C(m,k)C(k,n)\\ & =\sum_{k=0}^{m}(-1)^{k+m}C(m,k)C(k,n)\qquad,\qquad f(n)\delta_{mn}=f(m)\delta_{mn} \end{align*}

(0)-2

後のほうのみ示す。
\begin{align*} a^{n} & =\left(1+(a-1)\right)^{n}\\ & =\sum_{k=0}^{n}C(n,k)(a-1)^{k}\\ & =\sum_{k=0}^{n}\sum_{j=0}^{k}C(n,k)C(k,j)(-1)^{k-j}a^{j}\\ & =\sum_{j=0}^{n}\sum_{k=j}^{n}C(n,k)C(k,j)(-1)^{k-j}a^{j}\\ & =\sum_{j=0}^{n}\sum_{k=0}^{n}C(n,k)C(k,j)(-1)^{k-j}a^{j} \end{align*} これより、
\begin{align*} \delta_{nj} & =\sum_{k=0}^{n}C(n,k)C(k,j)(-1)^{k-j}\\ & =\frac{\left(-1\right)^{n}}{\left(-1\right)^{j}}\sum_{k=0}^{n}C(n,k)C(k,j)(-1)^{k-n}\\ & =\sum_{k=0}^{n}C(n,k)C(k,j)(-1)^{k+n} \end{align*}