扰动次数

扰动形式

扰动参数取值

生成函数

一次

${\alpha }_1\left(z^{-1}+z-2\right)+1$

${\alpha }_1=0$

$S=-\frac{{\left(1+z\right)}^4\left(1-4z+z^2\right)}{16z^3}$

二次

${\displaystyle \sum }_{k=1}^2{\alpha }_k\left(z^{-k}+z^k-2\right)+1$

${\alpha }_1=-4{\alpha }_2$

$S=-\frac{\begin{array}{l}{\left(1+z\right)}^4\left(1-4z+z^2\right)({\alpha }_2{z}^{-2}\\ -4{\alpha }_2{z}^{-1}+\left(1+6{\alpha }_2\right)-4{\alpha }_{2}z+{\alpha }_2{z}^2)\end{array}}{16z^3}$ ， ${\alpha }_2$ 为任意值

三次

${\displaystyle \sum }_{k=1}^3{\alpha }_k\left(z^{-k}+z^k-2\right)+1$

${\alpha }_1=-4{\alpha }_2-9{\alpha }_3$

$S=-\frac{\begin{array}{l}{\left(1+z\right)}^4\left(1-4z+z^2\right)({\alpha }_3{z}^{-3}+{\alpha }_2{z}^{-2}+\left(-4{\alpha }_2-9{\alpha }_3\right)z^{-1}\\ +\left(1+6{\alpha }_2+16{\alpha }_3\right)+\left(-4{\alpha }_2-9{\alpha }_3\right)z+{\alpha }_2{z}^2+{\alpha }_3{z}^3)\end{array}}{16z^{3b}}$ ， ${\alpha }_2,{\alpha }_3$ 为任意值