差分的性质

差分的性质

1)各阶差分均可用函数值表示

例如

$${\Delta }^n{f}_k=(\text{E}-\text{I}{)}^n{f}_k=\sum _{j=0}^n(-1{)}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right){\text{E}}^{n-j}{f}_k=\sum _{j=0}^n(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})f_{n+k-j}$$

$${\nabla }^n{f}_k=(1-{\mathrm{E}}^{-1}{)}^n{f}_k=\sum _{j=0}^n(-1{)}^{n-j}\left(\genfrac{}{}{0ex}{}{n}{j}\right){\mathrm{E}}^{j-n}{f}_k=\sum _{j=0}^n(-1{)}^{n-j}(\genfrac{}{}{0ex}{}{n}{j})f_{k+j-n}$$

其中$\left(\genfrac{}{}{0ex}{}{n}{j}\right)=\frac{n(n-1)\cdots (n-j+1)}{j!}$为二项式展开系数.

2)用各阶差分表示函数值

例如用向前差分表示$f_{n+k}$,即

$$f_{n+k}={\mathrm{E}}^n{f}_k=(\mathrm{I}+\Delta {)}^n{f}_k=\sum _{j=0}^n(\genfrac{}{}{0ex}{}{n}{j}){\Delta }^j{f}_k$$

3)差商与差分的关系

对于向前差分，有

$$f[x_k,x_{k+1}]=\frac{f_{k+1}-f_k}{x_{k+1}-x_k}=\frac{\Delta f_k}{h}$$

$$f[x_k,x_{k+1},x_{k+2}]=\frac{f[x_{k+1},x_{k+2}]-f[x_k,x_{k+1}]}{x_{k+2}-x_k}=\frac{1}{2h^2}{\Delta }^2{f}_k,$$

更一般的，

$$f[x_k,x_{k+1},\cdot \cdot \cdot ,x_{k+m}]=\frac{1}{m!}\frac{1}{h^m}{\Delta }^m{f}_k(m=1,2,\cdot \cdot \cdot ,n).$$

同理，对于向后差分，有

$$f[x_k,x_{k-1},\cdot \cdot \cdot ,x_{k-m}]=\frac{1}{m}!\frac{1}{h^m}{\nabla }^m{f}_k$$