$W_0(x)^r$的$n$阶导数

hbghlyj

$r\inN$, $W_{0}(x)^{r}$在0的$n(≥r)$阶导数为
\[-r\left(-n\right)^{n-r-1}\frac {n!}{(n-r)!}\]

hbghlyj

Formal power series
$g\in K[[X]]$为$f\in K[[X]]$的反函数, $f_0=g_0=0$, $f_{1}=g_{1}=1$, 则对于复数$\alpha,\beta$, 若$m=-\alpha -\beta \in \mathbb {N}$,
$${\frac {1}{\alpha }}[X^{m}]\left({\frac {f}{X}}\right)^{\alpha }=-{\frac {1}{\beta }}[X^{m}]\left({\frac {g}{X}}\right)^{\beta }.$$