# Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $$W$$ function as $$W(x)$$, I add as reference that Wikipedia has the article Lambert $$W$$ function. The particular value $$W(1)=\Omega$$ is known as the omega constant, see the Wikipedia Omega constant. We denote the imaginary unit as $$i=\sqrt{-1}$$.

Question. I would like to know if it is possible to deduce what about the irrationality or transcendence of each one of the following real constants $$i^{i\Omega}$$ and $$2^{\Omega}$$. Many thanks.

I'm asking what about the state of art, or what work can be done for my question. I don't know if my question is in the literature, if this is the case please refer the literature answering as a reference request and I try to search and read the statements from the literature. For the first question I tried a variant of Gelfond's constant, and for the second question I wondered about it, if I'm remember well and I'm right, in the context of an application of the six exponentials theorem.

I was inspired in my question posted on Mathematics Stack Exchange MSE 3579844 (asked Mar 13) that previously I've also asked on MathOverflow, and in the linked section of the Wikipedia's article with title List of unsolved problems in mathematics. As reference I know also the statement of Gelfond–Schneider theorem

• For the persons interested in this kind of questions, there is an excellent preprint on arXiv by Wadim Zudilin as arXiv:2004.11029 and title Diophantine problems related to the Omega constant – user142929 Jun 14 at 6:27
• Many thanks for the edit (I had not noticed the typo) @J.W.Tanner – user142929 Jun 15 at 6:13