Inverse of $f(x)=x^n(1-x)^k$

Question

I am trying to find an inverse of a function \begin{align} f(x)=x^n(1-x)^k, x \in (0,1) \end{align} where $n$ and $k$ are some positive integers.

I know that his function doesn't have a 'pure' inverse. However, it should have upper and lower branches.

Also, the inverse cannot be written in terms of elementary functions. Therefore, it has to be written in terms of some generalized functions like the Lambert-W function.

Question: Does $f$ have an inverse in terms of some generalized functions?

Answer 1

The Lagrange inversion formula gives $$a_i = \frac 1 {i!} \left. \frac {d^{i - 1}} {dy^{i - 1}} \left( \frac y {f(y)^{1/n}} \right)^{\! i} \, \right|_{y = 0} = \frac 1 {i!} \left( \frac {i k} n \right)_{\! i - 1}, \\ f^{-1}(y) = \sum_{i \geq 1} a_i y^{i/n} = y^{1/n} \hspace {1.5px} {_2 \hspace {-1px} \Psi_2} {\left( y^{1/n} \middle| {(1, 1), (\frac k n, 1 + \frac k n) \atop (2, 1), (\frac k n, \frac k n)} \right)},$$ where $(b)_i$ is the rising factorial and $\Psi$ is the Fox-Wright function. The second branch taking the values in $(0, 1)$ is $1 - f^{-1}(y)$ with $n$ and $k$ interchanged.