# 下降幂多项式

## 下降幂的定义

$$x$$$$n$$阶下降幂$$x^{\underline n}=\prod_0^{n-1}(x-i) = \frac{x!}{(x-n)!}$$

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### 快速求解$$x^{\underline n}$$的展开形式

$$x^{\underline{n}}=x(x-1)\cdots (x-n+1)$$

\begin{aligned} F(x-n)=\sum_{i=0}^{n} [x^i]F(x) \cdot (x-n)^i\end{aligned}

## FFP与其点值的$$\text{EGF}$$

\begin{aligned}EGF(F(x))=\sum_{i=0}^{\infty}\frac{x^i}{i!}\sum_{j=0}^{n} \frac{i!}{(i-j)!}\cdot F_j\end{aligned}

\begin{aligned}EGF(F(x))=\sum_{i=0}^{\infty}x^i \sum_{j=0}^{n} \frac{1}{(i-j)!}\cdot F_j\end{aligned}

\begin{aligned}EGF(F(x))=\sum_{i=0}^{n} F_i \sum_{j=i}^{\infty}\frac{1}{(j-i)!} x^j\end{aligned}

\begin{aligned}EGF(F(x))=\sum_{i=0}^{n} F_i \cdot x^i \sum_{j=0}^{\infty}\frac{1}{j!} x^j\end{aligned}

\begin{aligned}EGF(F(x))=\sum_{i=0}^{n} F_i \cdot x^i e^x\end{aligned}

Tips: $$e^x$$直接带入展开式\begin{aligned} e^{ax}=\sum_0^{\infty}\frac{(ax)^i}{i!} \end {aligned}

\begin{aligned}EGF(F(x))=\sum_{i=0}^{n} F_i \cdot x^ie^x\end{aligned}

\begin{aligned} F_i=\frac{EGF(F(x))}{x^ie^x} \end{aligned}

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## FFP卷积

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Tips: 下面的知识恐怕需要先学多点求值/快速插值

## 多项式转FFP

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## FFP转多项式

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## 关于上升幂

$$x^{\overline n}=\frac{(x+n-1)!}{(x-1)!}=x(x+1)(x+2)\cdots(x+n-1)$$

posted @ 2020-06-09 16:03  chasedeath  阅读(3763)  评论(0编辑  收藏  举报