The 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.
Over finite fields, the inverse of an exponential function is taken as discrete logarithm (given $a,g$ as elements of a cyclic group generated by $g$, the element $x$ in $g^x=a$ is the discrete logarithm of $a$).
Is it sensible to ask for definitions of the analogs of higher logarithms for $s>1$ over finite fields?