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

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.

• Many thanks for your attention and edit @Glorfindel Jun 18, 2020 at 15:02
• Just for fun, note that it also works for: $$\xi_i=\sqrt{-1+W\left(-1+\sqrt{-1+W(-1+\ldots)}\right)}\tag{1}$$ which yields the complex number $\xi_i =0.430438...+ 0.96479388...i$, that satisfies the identity: $$e^{\xi_i^2+1}=\frac{\xi_i-1}{\xi_i^2+1}$$
– Agno
Jun 18, 2020 at 23:29
• Many thanks your expressions/formulas always have a great mathematical beauty @Agno Jun 19, 2020 at 6:25

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.