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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.

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Q: Can the Lambert $W$ function be written as an inverse of a hypergeometric function?

A: $x=W(y)$ is the solution of $_1F_1(2;1;x-1)=y/e$.

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  • $\begingroup$ Its nice answer. thank you.I also want to ask if there is any relationships lambert w and gaussian hypergeometric function or not. Can you suggest any books or articles about it?@carlobeenakker $\endgroup$
    – queen28
    Commented Jun 18, 2021 at 19:33
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    $\begingroup$ $x=y^{-1}W(-e^{-y}y)$ is the solution to $_2F_1(1,1;2;x+1)=y$. $\endgroup$
    – Carlo Beenakker
    Commented Jun 19, 2021 at 9:23
  • $\begingroup$ woow that's great. thanks :) So do you have a reference? I know something about hypergeometric functions but i should study lambert w function:) @carlobeenakker $\endgroup$
    – queen28
    Commented Jun 19, 2021 at 17:40
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    $\begingroup$ it follows directly from $_2F_1(1,1;2;x)=-x^{-1}\ln(1-x)$ $\endgroup$
    – Carlo Beenakker
    Commented Jun 19, 2021 at 18:59

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