# Relationship between Lambert $W$ function and Hypergeometric function

The Lambert $$W$$ Function is defined in this Wikipedia entry, while the Hypergeometric Function is defined in this other Wikipedia entry. There exists also a multivariate generalization which solves the following equation $$e^{-c x}=d \frac{\left(x-a_{0}\right)\left(x-a_{1}\right) \cdots\left(x-a_{n}\right)}{\left(x-b_{0}\right)\left(x-b_{1}\right) \cdots\left(x-b_{m}\right)}$$

as I read from Quora post. This equation has some analogies in Hypergeometric functions as well.

I also would like to know if the Lambert $$W$$ Function can be written as an inverse of Hypergeometric functions: is it so? Or are there any other kind of relationship about them? Thanks for your answers and references.

Q: Can the Lambert $$W$$ function be written as an inverse of a hypergeometric function?
A: $$x=W(y)$$ is the solution of $$_1F_1(2;1;x-1)=y/e$$.
• $x=y^{-1}W(-e^{-y}y)$ is the solution to $_2F_1(1,1;2;x+1)=y$. Jun 19, 2021 at 9:23
• it follows directly from $_2F_1(1,1;2;x)=-x^{-1}\ln(1-x)$ Jun 19, 2021 at 18:59