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/ |
| SNSボタン |
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(上昇階乗・下降階乗)の１項間漸化式
\[
P(x+1,y)=\frac{x+1}{x-y+1}P(x,y)
\]
階乗冪(下降階乗・上昇階乗)の差分
\[
P(x,y)=\frac{1}{y+1}\left(P(x+1,y+1)-P(x,y+1)\right)
\]
階乗冪(下降階乗・上昇階乗)の1/2値
\[
P\left(-\frac{1}{2},n\right)=\frac{(-1)^{n}(2n-1)!}{2^{2n-1}(n-1)!}
\]