多項係数の畳み込み

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ヴァンデルモンドの恒等式みたいな畳み込みは、多項係数でも成り立つのか試したらうまくいきました。
証明も出来たので見てください。

まず下降冪多項定理を証明します。
$(x)_{n}$ を下降冪とします。

$(x_{0} + x_{1} + x_{2} + \dots + x_{r})_{n + 1} = (x_{0} + x_{1} + x_{2} + \dots + x_{r} - n) (x_{0} + x_{1} + x_{2} + \dots + x_{r})_{n}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} ((x_{0} - y_{0}) + (x_{1} - y_{1}) + (x_{2} - y_{2}) + \dots + (x_{r} - y_{r})) \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0} - y_{0}) + (x_{1} - y_{1}) + (x_{2} - y_{2}) + \dots + (x_{r} - y_{r})) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0} - y_{0}) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}} + (x_{1} - y_{1}) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}} + (x_{2} - y_{2}) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}} + \dots + (x_{r} - y_{r}) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}})$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0})_{y_{0} + 1} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}} + (x_{0})_{y_{0}} (x_{1})_{y_{1} + 1} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}} + (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2} + 1} \dots (x_{r})_{y_{r}} + \dots + (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r} + 1})$
$= \sum_{(y_{0} + 1) + y_{1} + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ (y_{0} - 1) y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$+ \sum_{y_{0} + (y_{1} + 1) + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} (y_{1} - 1) y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$+ \sum_{y_{0} + y_{1} + (y_{2} + 1) + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} (y_{2} - 1) \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$\dots + \sum_{y_{0} + y_{1} + y_{2} + \dots + (y_{r} + 1) = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots (y_{r} - 1) \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n + 1} (\begin{array}{r} (\begin{array}{cc} n \\ (y_{0} - 1) y_{1} y_{2} \dots y_{r} \end{array}) \end{array}$ $+ \begin{array}{r} (\begin{array}{cc} n \\ y_{0} (y_{1} - 1) y_{2} \dots y_{r} \end{array}) \end{array} +$ $\begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} (y_{2} - 1) \dots y_{r} \end{array}) \end{array} + \dots$ $+ \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots (y_{r} - 1) \end{array}) \end{array}) (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n + 1 \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$

これを踏まえてさらに進みます。

1.1

$(x_{0} + x_{1} + x_{2} + \dots + x_{r})_{n} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r})!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - n!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} \frac{x_{0}!}{x_{0} - y_{0}!} \frac{x_{1}!}{x_{1} - y_{1}!} \frac{x_{2}!}{x_{2} - y_{2}!} \dots \frac{x_{r}!}{x_{r} - y_{r}!}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r})!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - n! n!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \frac{x_{0}!}{x_{0} - y_{0}! y_{0}!} \frac{x_{1}!}{x_{1} - y_{1}! y_{1}!} \frac{x_{2}!}{x_{2} - y_{2}! y_{2}!} \dots \frac{x_{r}!}{x_{r} - y_{r}! y_{r}!}$

$\begin{array}{r} (\begin{array}{cc} x_{0} x_{1} x_{2} \dots x_{r} \\ n \end{array}) \end{array}$
$＝ \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} x_{0} \\ y_{0} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} x_{1} \\ y_{1} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} x_{2} \\ y_{2} \end{array}) \end{array} \dots$ $\begin{array}{r} (\begin{array}{cc} x_{r} \\ y_{r} \end{array}) \end{array}$

1.2

$(x_{0} + x_{1} + x_{2} + \dots + x_{r})_{n} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})_{y_{0}} (x_{1})_{y_{1}} (x_{2})_{y_{2}} \dots (x_{r})_{y_{r}}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r})!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - n!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} \frac{x_{0}!}{x_{0} - y_{0}!} \frac{x_{1}!}{x_{1} - y_{1}!} \frac{x_{2}!}{x_{2} - y_{2}!} \dots \frac{x_{r}!}{x_{r} - y_{r}!}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r})!}{x_{0}! x_{1}! x_{2}! \dots x_{r}!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} \frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - n!}{(x_{0} - y_{0})! (x_{1} - y_{1})! (x_{2} - y_{2})! \dots (x_{r} - y_{r})!}$

$(x_{0} + x_{1} + x_{2} + \dots + x_{r}) ＝ r$ とすると

$\begin{array}{r} (\begin{array}{cc} r \\ x_{0} x_{1} x_{2} \dots x_{r} \end{array}) \end{array}$ $＝ \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} r - n \\ (x_{0} - y_{0}) (x_{1} - y_{1}) (x_{2} - y_{2}) \dots (x_{r} - y_{r}) \end{array}) \end{array}$

$(x)^{n}$ を上昇冪とします。

$(x_{0} + x_{1} + x_{2} + \dots + x_{r})^{n + 1} = (x_{0} + x_{1} + x_{2} + \dots + x_{r} + n) (x_{0} + x_{1} + x_{2} + \dots + x_{r})^{n}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} ((x_{0} + y_{0}) + (x_{1} + y_{1}) + (x_{2} + y_{2}) + \dots + (x_{r} + y_{r})) \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0} + y_{0}) + (x_{1} + y_{1}) + (x_{2} + y_{2}) + \dots + (x_{r} + y_{r})) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0} + y_{0}) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}} + (x_{1} + y_{1}) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}} + (x_{2} + y_{2}) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}} + \dots + (x_{r} + y_{r}) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}})$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} ((x_{0})^{y_{0} + 1} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}} + (x_{0})^{y_{0}} (x_{1})^{y_{1} + 1} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}} + (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2} + 1} \dots (x_{r})^{y_{r}} + \dots + (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r} + 1})$
$= \sum_{(y_{0} + 1) + y_{1} + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ (y_{0} - 1) y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$+ \sum_{y_{0} + (y_{1} + 1) + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} (y_{1} - 1) y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$+ \sum_{y_{0} + y_{1} + (y_{2} + 1) + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} (y_{2} - 1) \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$\dots + \sum_{y_{0} + y_{1} + y_{2} + \dots + (y_{r} + 1) = n + 1} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots (y_{r} - 1) \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n + 1} (\begin{array}{r} (\begin{array}{cc} n \\ (y_{0} - 1) y_{1} y_{2} \dots y_{r} \end{array}) \end{array}$ $+ \begin{array}{r} (\begin{array}{cc} n \\ y_{0} (y_{1} - 1) y_{2} \dots y_{r} \end{array}) \end{array} +$ $\begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} (y_{2} - 1) \dots y_{r} \end{array}) \end{array} + \dots$ $+ \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots (y_{r} - 1) \end{array}) \end{array}) (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$= \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n + 1} \begin{array}{r} (\begin{array}{cc} n + 1 \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$

2.1

$(x_{0} + x_{1} + x_{2} + \dots + x_{r})^{n} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} (x_{0})^{y_{0}} (x_{1})^{y_{1}} (x_{2})^{y_{2}} \dots (x_{r})^{y_{r}}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) + n - 1!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - 1!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} \frac{(x_{0} + y_{0} - 1)!}{x_{0} - 1!} \frac{(x_{1} + y_{1} - 1)!}{x_{1} - 1!} \frac{(x_{2} + y_{2} - 1)!}{x_{2} - 1!} \dots \frac{(x_{r} + y_{r} - 1)!}{x_{r} - 1!}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) + n - 1!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - 1!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} n \\ y_{0} y_{1} y_{2} \dots y_{r} \end{array}) \end{array} \frac{(x_{0} + y_{0} - 1)!}{x_{0} - 1!} \frac{(x_{1} + y_{1} - 1)!}{x_{1} - 1!} \frac{(x_{2} + y_{2} - 1)!}{x_{2} - 1!} \dots \frac{(x_{r} + y_{r} - 1)!}{x_{r} - 1!}$
$\frac{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) + n - 1!}{(x_{0} + x_{1} + x_{2} + \dots + x_{r}) - 1! n!} = \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \frac{(x_{0} + y_{0} - 1)!}{x_{0} - 1! y_{0}!} \frac{(x_{1} + y_{1} - 1)!}{x_{1} - 1! y_{1}!} \frac{(x_{2} + y_{2} - 1)!}{x_{2} - 1! y_{2}!} \dots \frac{(x_{r} + y_{r} - 1)!}{x_{r} - 1! y_{r}!}$

$\begin{array}{r} (\begin{array}{cc} x_{0} + x_{1} + x_{2} + \dots + x_{r} + n - 1 \\ n \end{array}) \end{array}$ $＝ \sum_{y_{0} + y_{1} + y_{2} + \dots + y_{r} = n} \begin{array}{r} (\begin{array}{cc} x_{0} + y_{0} - 1 \\ y_{0} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} x_{1} + y_{1} - 1 \\ y_{1} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} x_{2} + y_{2} - 1 \\ y_{2} \end{array}) \end{array} \dots$ $\begin{array}{r} (\begin{array}{cc} x_{r} + y_{r} - 1 \\ y_{r} \end{array}) \end{array}$

投稿日：2024年6月11日

更新日：2024年7月5日

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多項係数の畳み込み