The Lambert $W$ Function is defined in [this Wikipedia entry](https://en.wikipedia.org/wiki/Lambert_W_function), while the Hypergeometric Function is defined in [this other Wikipedia entry](https://en.wikipedia.org/wiki/Hypergeometric_function). 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](https://www.quora.com/How-is-the-Lambert-W-function-derived). 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.



Relationship between Lambert $W$ function and Hypergeometric function
**Q:** Can the Lambert $W$ function be written as an inverse of a hypergeometric function?


**A:** $x=W(y)$ is the solution of $\_1F\_1(2;1;x-1)=y/e$.


