階乗冪

階乗冪(上昇階乗・下降階乗)の１項間漸化式

by nomura · 2020年6月11日

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(1)

P (x + 1, y) = \frac{x + 1}{x - y + 1} P (x, y)

(2)

P (x - 1, y) = \frac{x - y}{x} P (x, y)

(3)

P (x, y + 1) = (x - y) P (x, y)

(4)

P (x, y - 1) = \frac{1}{x - y + 1} P (x, y)

(5)

Q (x + 1, y) = \frac{x + y}{x} Q (x, y)

(6)

Q (x - 1, y) = \frac{x - 1}{x + y - 1} Q (x, y)

(7)

Q (x, y + 1) = (x + y) Q (x, y)

(8)

Q (x, y - 1) = \frac{1}{x + y - 1} Q (x, y)

(1)

\begin{aligned} P (x + 1, y) & = \frac{(x + 1)!}{(x + 1 - y)!} \\ = \frac{x + 1}{x + 1 - y} \frac{x!}{(x - y)!} \\ = \frac{x + 1}{x - y + 1} P (x, y) \end{aligned}

(2)

\begin{aligned} P (x - 1, y) & = \frac{(x - 1)!}{(x - 1 - y)!} \\ = \frac{x - y}{x} \frac{x!}{(x - y)!} \\ = \frac{x - y}{x} P (x, y) \end{aligned}

(3)

\begin{aligned} P (x, y + 1) & = \frac{x!}{(x - y - 1)!} \\ = (x - y) \frac{x!}{(x - y)!} \\ = (x - y) P (x, y) \end{aligned}

(4)

\begin{aligned} P (x, y - 1) & = \frac{x!}{(x - y + 1)!} \\ = \frac{1}{x - y + 1} \frac{x!}{(x - y)!} \\ = \frac{1}{x - y + 1} P (x, y) \end{aligned}

(5)

\begin{aligned} Q (x + 1, y) & = \frac{(x + 1 + y - 1)!}{(x + 1 - 1)!} \\ = \frac{x + y}{x} \frac{(x + y - 1)!}{(x - 1)!} \\ = \frac{x + y}{x} Q (x, y) \end{aligned}

(6)

\begin{aligned} Q (x - 1, y) & = \frac{(x - 1 + y - 1)!}{(x - 1 - 1)!} \\ = \frac{x - 1}{x + y - 1} \frac{(x + y - 1)!}{(x - 1)!} \\ = \frac{x - 1}{x + y - 1} Q (x, y) \end{aligned}

(7)

\begin{aligned} Q (x, y + 1) & = \frac{(x + y + 1 - 1)!}{(x - 1)!} \\ = (x + y) \frac{(x + y - 1)!}{(x - 1)!} \\ = (x + y) Q (x, y) \end{aligned}

(8)

\begin{aligned} Q (x, y - 1) & = \frac{(x + y - 1 - 1)!}{(x - 1)!} \\ = \frac{1}{x + y - 1} \frac{(x + y - 1)!}{(x - 1)!} \\ = \frac{1}{x + y - 1} Q (x, y) \end{aligned}

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タイトル	階乗冪(上昇階乗・下降階乗)の１項間漸化式
URL	https://www.nomuramath.com/n0lg5oq0/
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階乗冪(上昇階乗・下降階乗)同士の関係

P (x, y) = P^{- 1} (x - y, - y)

階乗冪(上昇階乗・下降階乗)とその逆数の値が0となるとき

\forall m, n \in Z, 0 \leq m < n \Leftrightarrow P (m, n) = 0

階乗冪(下降階乗・上昇階乗)の和分

\sum_{k = 1}^{m} P (k, n) = \frac{1}{n + 1} P (m + 1, n + 1)

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)