Th polylogarithm function is defined by $$Li_s(z)=\sum_{k=1}^\infty\frac{z^k}{k^s}.$$ At $s=1$, we have the natural logarithm function. We have the inverse of natural logarithm function as the exponential function.
What are the analogous inverses of higher logarithms for $s>1$?
$$\sum_1^\infty\frac{z^k}{k}=-\ln(1-z)\iff\sum_1^\infty\frac{z^k}{k^n}=-\left(\int\frac1z\right)^{n-1}\ln(1-z)dz^{n-1}\iff$$
$$-\ln(1-z)=\left(z\frac d{dz}\right)^{n-1}\text{Li}_nz\iff z=1-\exp\left[-\left(z\frac d{dz}\right)^{n-1}\text{Li}_nz\right].$$
For $n\not\in\mathbb N$, see fractional calculus.