階乗冪(下降階乗・上昇階乗)の和分
(1)
\(m\in\mathbb{N}_{0}\cup\{-1\}\;\land\;n\in\mathbb{Z}\setminus\{-1\}\)\[ \sum_{k=1}^{m}P(k,n)=\frac{1}{n+1}P(m+1,n+1) \]
(2)
\(m-n\in\mathbb{N}_{0}\cup\{-1\}\;\land\;n\in\mathbb{Z}\setminus\mathbb{N}\)\[ \sum_{k=1}^{m}P^{-1}(k,n)=\frac{1}{n-1}\left((1-n)!-P(m-n+1,1-n)\right) \]
(3)
\(m+n\in\mathbb{N}_{0}\;\land\;n\in\mathbb{Z}\setminus\{-1\}\)\[ \sum_{k=1}^{m}Q(k,n)=\frac{1}{n+1}Q(m,n+1) \]
(4)
\(m\in\mathbb{N}_{0}\;\land\;n\in\mathbb{Z}\setminus\{1\}\)\[ \sum_{k=1}^{m}Q^{-1}(k,n)=\frac{1}{n-1}\left(\frac{1}{(n-1)!}-\frac{1}{Q(m+1,n-1)}\right) \]
(1)
\begin{align*} \sum_{k=0}^{m}P(k,n) & =\sum_{k=0}^{m}\left\{ \frac{1}{n+1}\left(P(k+1,n+1)-P(k,n+1)\right)\right\} \\ & =\frac{1}{n+1}\left(P(m+1,n+1)-P(0,n+1)\right)\\ & =\frac{1}{n+1}P(m+1,n+1) \end{align*}(2)
\begin{align*} \sum_{k=1}^{m}P^{-1}(k,n) & =\sum_{k=1}^{m}\frac{1}{n-1}\left(\frac{1}{P(k-1,n-1)}-\frac{1}{P(k,n-1)}\right)\\ & =\frac{1}{n-1}\left(\frac{1}{P(0,n-1)}-\frac{1}{P(m,n-1)}\right)\\ & =\frac{1}{n-1}\left((1-n)!-P(m-n+1,1-n)\right) \end{align*}(3)
\begin{align*} \sum_{k=1}^{m}Q(k,n) & =\sum_{k=1}^{m}\frac{1}{n+1}\left(Q(k,n+1)-Q(k-1,n+1)\right)\\ & =\frac{1}{n+1}\left(Q(m,n+1)-Q(0,n+1)\right)\\ & =\frac{1}{n+1}Q(m,n+1) \end{align*}(4)
\begin{align*} \sum_{k=1}^{m}Q^{-1}(k,n) & =\sum_{k=1}^{m}\frac{1}{n-1}\left(\frac{1}{Q(k,n-1)}-\frac{1}{Q(k+1,n-1)}\right)\\ & =\frac{1}{n-1}\left(\frac{1}{Q(1,n-1)}-\frac{1}{Q(m+1,n-1)}\right)\\ & =\frac{1}{n-1}\left(\frac{1}{(n-1)!}-\frac{1}{Q(m+1,n-1)}\right) \end{align*}ページ情報
| タイトル | 階乗冪(下降階乗・上昇階乗)の和分 |
| URL | https://www.nomuramath.com/jpiqxpwt/ |
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階乗冪(上昇階乗・下降階乗)の母関数
\[
\sum_{k=0}^{\infty}P(k,n)x^{k}=\frac{x^{n}n!}{(1-x)^{n+1}}
\]
階乗冪(下降階乗・上昇階乗)の微分
\[
\frac{d}{dx}P(x,y) =P(x,y)\left\{ \psi(1+x)-\psi(1+x-y)\right\}
\]
和の階乗冪(下降階乗・上昇階乗)
\[
P(x+y,n)=\sum_{k=0}^{n}C(n,k)P(x,k)P(y,n-k)
\]
階乗冪(上昇階乗・下降階乗)の定義
\[
P(x,y)=\frac{x!}{(x-y)!}
\]