Until recently, I had the arrogance to believe that I was the first to obtain an infinite series for the Omega constant ( also known as W(1), where W is the Lambert W function). I obtained the infinite series using Whittaker’s root series formula. You can see more details by reading the page for the OEIS sequence A370490 and A370491.
I found out that there was already a way to obtain an infinite series for the Omega by reading the Lambert W function Wikipedia page. The page mentions in a few places the Lagrange inversion theorem , so I clicked on the link that brings us to the Wikipedia page dedicated to theorem. The Lagrange Inversion Theorem page in turn has a section dedicated to Lambert W function that has the following power series.
W(1) can be obtained when z=-1. If you need more terms for the expansion see OEIS sequence A274447 and A274448.
More from Whittaker’s root series formula
I obtained my infinite series by applying Whittaker’s root series formula on the Taylor series expansion of -x + e^(-x). The idea to apply the formula to this specific Taylor series expansion was sparked by the fact that e^(-Ω)=Ω.
The most well known Omega constant identity is Ωe^Ω=1. So a completely new infinite series for Ω can probably be obtained by applying Whittaker’s root series formula on the Taylor series expansion of -1+ xe^x.
More Love for Lambert W Function and Omega Constant
Lambert W function seems to have a lot of applications in physics, biology, ecology and probability. Of course , the Lambert W function also has a few applications in mathematics. Most of these applications are covered in the book “The Lambert W Function” by István Mező ( I have yet to buy the book myself).
The Omega constant is also very close to a constant related to conjectures about the gap of consecutive prime numbers. See OEIS sequence A038458 for more details.
Overall, I believe that more people should be aware of the Omega constant or the Lambert W function. In the future we may find out that they appear in other interesting places. If nothing else, maybe I stirred some interest about the Lagrange Inversion Theorem or Whittaker’s root series formula.