Define the real number $\xi$ satisfying $$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia has the article with title Lambert $W$ function. Then $(1)$ is similar to the well-known representation for the golden ratio $(1+\sqrt{5})/2$, where were the Lambert $W$ function appears in an alternating way instead of the corresponding nested square root, see the subsection 4.4 Alternative forms from the Wikipedia Golden ratio, or [1] in Spanish). From this we conclude the following easy statement.
Claim. One has that $\xi\approx 1.3918$ satisfies the identity $$e^{\xi^2-1}=\frac{1}{\xi-1},\tag{2}$$ and thus $\xi$ is transcendental.
Proof sketch. As usual we deduce $\xi^2-1=W(1+\xi)$ from $(1)$, and calculating with the inverse $W^{-1}(x)$ we conclude $(2)$. By contradiction we prove the transcendence of $\xi$, as an application of Lindemann-Wierstrass theorem (see reference [2]).$\square$
I don't know if my claim is in the literature or if next questions are in the literature, if this post have a good mathematical content and is on topic, please feel free to refer the literature answering my questions as a reference request that I'm going to search and read those statements from the literature.
Question 1. The encyclopedia Wolfram MathWorld have an article with title e Continued Fraction. My purpose to write the post was to know how a mathematician tries to take advantage from a simple claim (previous) to deduce more related and advanced statements. Can you show any continued fraction representation related to our real number $\xi$ (as you see the examples of Wolfram MathWorld for $e$ are diverse/varied)? Many thanks
I wanted to ask previous question as an invitation and with the purpose to learn what can be a good/interesting continued fraction or issue concerning continued fractions, associated to our real number $\xi$. Next question also is in this spirit, I don't know if it can be made some work about it.
Question 2. I would like to know/determine some (wider) class of functions $f(x)$ for which I can to repeat my experiment/claim and state that the real number $\xi_f$ defined as $\xi_f=\sqrt{1+f\left(1+\sqrt{1+f(1+\ldots)}\right)}$ will be a real transcendental number. Many thanks.
I mean with this second question a case study of what conditions are required for the real functions $f(x)$ with the purpose to get an extension of my Claim (if this is in the literature feel free to answer this second question as a reference request). I'm asking what work can be done for an extension of my claim (thus an example of a wider and suitable class of functions $f(x)$). Isn't required to find nice-closed forms, just decide what is a suitable class of functions for which I can to prove transcendence.
References:
[1] Samuel G. Moreno and Esther M. García Caballero, Uno, dos y $\ldots\,$¡$\phi$!, Miniaturas matemáticas de La Gaceta de la RSME, La Gaceta de la Real Sociedad Matemática Española, Vol. 20 (2017), Núm. 1, Pág. 170.
[2] The section Numbers proven to be transcendental from the Wikipedia Transcendental number.