On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio

Question

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.

Answer 1

A quick thought for your Question 2.

If the limit for $\xi_f$ exists, then as you noted we'll get $\xi_f ^2 - 1 = f(\xi_f +1)$. If we define $g(x) = f(x+1) - x^2 +1 = f(x+1) - (x+1)^2 + 2(x+1)$, then we have $x^2 - 1 = f(x+1)$ iff $g(x) = 0$.

So your second question more or less reduces to saying ``we have a function $g$, and we would like to know when the equation $g(x) = 0$ forces $x$ to be transcendental."

Or perhaps a bit more to the point would be to define $h(x) = \sqrt{1+f(x)}$. Then you are wondering about fixed points of $h$, and you're hoping there's only one.

I'm afraid that question at the moment is too broad to really sink your teeth into (or to make much progress) since it seems to be the same as asking "which fixed points are transcendental." But I certainly like the creativity of the idea!