I know that there are some open problems concerning the irrationality and trancescende of certain combinations (suitable expressions as sums/differences, products/quotients and exponentiations/logarithms) of the constants $\pi$ and $e$, I know it (I add the following reference) from the section Analysis of the Wikipedia List of unsolved problems in mathematics.
In this post we denote the value at $x=1$ of the main/principal branch of the Lambert $W$ function as $\Omega=W(1)$. The Wikipedia encyclopedia has the article see this Wikipedia Omega constant.
Question. I would like to know what can be a good proposal(s) of open problem concerning the transcendence or irrationality, at research level, for suitable combinations of the real numbers $\pi$, $e$ and $\Omega=W(1)$. Many thanks.
The only requirement is that your expression must involve the omega constant (expressions also involving the $\Omega$ and $\pi$ or $e$ are desirable). Isn't required state your proposal(s) as a conjecture, just as an interesting open problem.
If you are able to create an example at research level for which you can prove, using your knowledges, that your expression is rational, algebraic irrational, or transcendental, I think that it can be also an answer for my question.
If there are such proposals of open problems in the literature, please refer the literature answering my question as a reference request and I try to search and read those from the literature.
Best Answer
I will suggest some (maybe) interesting constants for which irrationality or transcendence can be studied. First let me recall some well known constants (this list could also be useful to formulate some other conjectures here):
$$G:=\frac{\Gamma^2 (1/4)}{2 \sqrt{2 \pi^3} }$$
known as Gauss's constant, which is transcendent. Then, we can define the following two lemniscate constants, which are transcendental:
$$ L_1:=\pi G $$
$$L_2 := \frac{1}{2 G}$$ (they both arise when looking for a formula for the arc lenght of a lemniscate).
Another interesting number, which arises in a similar way to the golden ratio, is the plastic constant $\rho$, which is the unique real solution of the equation $x^3=x+1$. This constant is irrational.
A well known number, which has been proven to be irrational by Apéry, is $\zeta(3)$, which is also called Apéry's constant.
Now we can study the irrationality/transcendence of the following numbers:
$$ \Omega + \pi, \, \Omega + G, \, \Omega^{\rho}, \, \Omega^{\zeta(3)}, \, \Omega e^{L_1}, \Omega^{\ln \pi}, \sqrt{2 \Omega}, ... $$
This list can become really long: you just have to combine in many other ways all the above constants with $\Omega$, $e$, $\pi$, $i$, ...