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

(0)-2

\begin{align*} \sum_{k=0}^{m}\left(-1\right)^{k+m}C\left(m,k\right)C\left(k,n\right) & =\sum_{k=0}^{m}\left(-1\right)^{k+m}C\left(m,n\right)C\left(m-n,m-k\right)\cmt{\because C\left(m,k\right)C\left(k,n\right)=C\left(m,n\right)C\left(m-n,m-k\right)}\\ & =C\left(m,n\right)\left(-1\right)^{m}\sum_{k=0}^{m}C\left(m-n,m-k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\left(-1\right)^{m}\sum_{k=0}^{m}C\left(m-n,k\right)\left(-1\right)^{m-k}\cmt{k\rightarrow m-k}\\ & =C\left(m,n\right)\sum_{k=0}^{m}C\left(m-n,k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\sum_{k=0}^{m-n}C\left(m-n,k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\left(1-1\right)^{m-n}\\ & =C\left(m,n\right)0^{m-n}\\ & =C\left(m,n\right)\delta_{m,n}\\ & =\delta_{m,n} \end{align*} \begin{align*} \sum_{k=0}^{m}\left(-1\right)^{k+m}C\left(m,k\right)C\left(k,n\right) & =\sum_{k=0}^{m}\left(-1\right)^{k+m}C\left(m,n\right)C\left(m-n,m-k\right)\cmt{\because C\left(m,k\right)C\left(k,n\right)=C\left(m,n\right)C\left(m-n,m-k\right)}\\ & =C\left(m,n\right)\left(-1\right)^{m}\sum_{k=0}^{m}C\left(m-n,m-k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\left(-1\right)^{m}\sum_{k=0}^{m}C\left(m-n,k\right)\left(-1\right)^{m-k}\cmt{k\rightarrow m-k}\\ & =C\left(m,n\right)\sum_{k=0}^{m}C\left(m-n,k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\sum_{k=0}^{m-n}C\left(m-n,k\right)\left(-1\right)^{k}\\ & =C\left(m,n\right)\left(1-1\right)^{m-n}\\ & =C\left(m,n\right)0^{m-n}\\ & =C\left(m,n\right)\delta_{m,n}\\ & =\delta_{m,n} \end{align*}

(0)-3

2つ目のみ示す。
\begin{align*} a^{n} & =\left(1+\left(a-1\right)\right)^{n}\\ & =\sum_{k=0}^{n}C\left(n,k\right)\left(a-1\right)^{k}\\ & =\sum_{k=0}^{n}\sum_{j=0}^{k}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k-j}a^{j}\\ & =\sum_{j=0}^{n}\sum_{k=j}^{n}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k-j}a^{j}\\ & =\sum_{j=0}^{n}\sum_{k=0}^{n}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k-j}a^{j} \end{align*} これより、
\begin{align*} \delta_{nj} & =\sum_{k=0}^{n}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k-j}\\ & =\frac{\left(-1\right)^{n}}{\left(-1\right)^{j}}\sum_{k=0}^{n}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k-n}\\ & =\sum_{k=0}^{n}C\left(n,k\right)C\left(k,j\right)\left(-1\right)^{k+n} \end{align*}