5
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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

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

$\endgroup$
4
  • $\begingroup$ 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
    Jun 18, 2021 at 19:33
  • 1
    $\begingroup$ $x=y^{-1}W(-e^{-y}y)$ is the solution to $_2F_1(1,1;2;x+1)=y$. $\endgroup$ 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
    Jun 19, 2021 at 17:40
  • 1
    $\begingroup$ it follows directly from $_2F_1(1,1;2;x)=-x^{-1}\ln(1-x)$ $\endgroup$ Jun 19, 2021 at 18:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.