数列变换

加和原理

$\sum_{n \geq 0} (f_{n} + g_{n}) x^{n} = F (z) + G (z)$

数列和等于函数和, 收敛域依赖于小的那个函数.

位移原理

Hadamard 积

两个数列的乘积称为 Hadamard 积
$\displaystyle{\sum_{n\geq 0}f_{n}g_{n}z^{n}:=(F\odot G)(z)={\frac {1}{2\pi }}\int_{0}^{2\pi }F\left({\sqrt {z}}e^{\imath t}\right)G\left({\sqrt {z}}e^{-\imath t}\right)\mathrm{d}t}$

等比原理

$\sum_{n \geq 1} \frac{f_{n}}{k^{n}} z^{n} = \sum_{n \geq 1} f_{n} \frac{z^{n}}{k^{n}} = F (\frac{z}{k})$

Zeta 变换

Zeta 主变换

定义 $\displaystyle\left\{ $\begin{matrix} k + 2 \\ j \end{matrix}$ \right\}_\ast:={\frac {1}{j!}}\times \sum_{m=1}^{j}{\binom {j}{m}}{\frac {(-1)^{j-m}}{m^k}}$

那么数列$f_n$ 的 Zeta 变换可以记为:

$\sum_{n \geq 1} \frac{f_{n}}{n^{k}} z^{n} = \sum_{j \geq 1} {\begin{matrix} k + 2 \\ j \end{matrix}}_{*} z^{j} F^{(j)} (z)$

where Then for $k \in \mathbb{Z}^{+}$ and some prescribed OGF, ${\displaystyle F(z)\in C^{\infty }}$ , i.e., so that the higher-order $j^{th}$ derivatives of$F(z)$ exist for all $j\geq 0$ , we have that

Zeta 逆变换

$\sum_{n \geq 0} n^{k} f_{n} z^{n} = \sum_{j = 0}^{k} {\begin{matrix} k \\ j \end{matrix}} z^{j} F^{(j)} (z)$

Polylogarithm 级数

提取原理

奇偶提取

$\begin{aligned} \sum_{n \geq 0} f_{2 n} z^{2 n} & = \frac{1}{2} (F (z) + F (- z)) \\ \sum_{n \geq 0} f_{2 n + 1} z^{2 n + 1} & = \frac{1}{2} (F (z) - F (- z)) \end{aligned}$

五级标题

六级标题

Borel 变换

Borel 变换描述了OGF与EGF间的转换关系.

$\begin{aligned} F (z) & = \sum_{n \geq 0} f_{n} z^{n} = \int_{0}^{\infty} \hat{F} (t z) e^{- t} d t \\ \hat{F} (z) & = \sum_{n \geq 0} \frac{f_{n}}{n!} z^{n} = \frac{1}{2 π} \int_{- π}^{π} F (z e^{- ı ϑ}) e^{e^{ı ϑ}} d ϑ \end{aligned}$

函数变换

加和原理

$\sum_{n \geq 0} (f_{n} + g_{n}) x^{n} = F (z) + G (z)$

Cauchy 积

$\sum (f * g) (n) := (\sum_{n = 0}^{\infty} f_{n} x^{n}) \cdot (\sum_{n = 0}^{\infty} g_{n} x^{n}) = \sum_{k = 0}^{\infty} h_{n} x^{n}$

where $\displaystyle h_{k}=\sum_{l=0}^{k}f_{l}g_{k-l}$ .

Faá di Bruno 复合

$\hat{H} (z) := \hat{F} (\hat{G} (z)) = \sum_{n = 0}^{\infty} \frac{h_{n}}{n!} z^{n}$

where: $\displaystyle h_n=\sum_{1\leq k\leq n}f_k\cdot B_{n,k}(g_1,g_2,\cdots,g_{n-k+1})+f_0\cdot\delta_{n,0}$

取自幂

$F (z)^{m} = f_{0}^{m} + \sum_{n \geq 1} (\sum_{1 \leq k \leq n} (m)_{k} f_{0}^{m - k} B_{n, k} (f_{1} \cdot 1!, f_{2} \cdot 2!, \dots)) \frac{z^{n}}{n!}$

取对数

$\log F (z) = \sum_{n \geq 1} (\sum_{1 \leq k \leq n} (- 1)^{k - 1} (k - 1)! B_{n, k} (f_{1} \cdot 1!, f_{2} \cdot 2!, \dots)) \frac{z^{n}}{n!}$

https://en.wikipedia.org/wiki/Generating_function_transformation