False formula of Galois and Abel fractal and the quintic

Question

6Well all start with the well-know sentence due to Abel : "At 16 years old i found a false formula for the general quintic "

After many attempts I found by myself this :

Fractal formula :

It's well know (but not for me since now) that we can use Newton's method to find an approximation of polynomials's roots (see quote Gauss)

The problem is how to start :

We can fasely tackle the problem in using iterative method :

An example :

Let the polynomial :

$$x^{5}+5x^{4}+8x^{3}+4x^{2}-3x-2$$

We want :

$$x^{5}+5x^{4}+8x^{3}+4x^{2}-3x-2\simeq 0$$

Or :

$$x^{2}\left(1+x+\frac{1}{x+1}+\frac{1}{1+\frac{1}{x+1}}\right)-1\simeq 0$$

A way to find an approximation of one of the roots is :

$$x=\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdot\cdot\cdot}}}}}+\frac{1}{1+\frac{1}{1+\frac{1}{\sqrt{1+\frac{1}{\sqrt{1+\cdot\cdot\cdot}}}}}}}}$$

Where we have fasely iterated the solution

Question :

How to formalize the exact solution in my example which seems bigger than all because in move ?

Perhaps a start is an infinite matrix...see also Nested root integral $\int_0^1 \frac{dx}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$

All helps is very welcome !

Answer 1

I finally find an answer with the concept of branched continued fraction for a reference see https://arxiv.org/pdf/1807.03271.pdf

Reference :

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivity

Mathias Pétréolle, Alan D. Sokal, Bao-Xuan Zhu

Edit : there was a similar approach as mine using cf see https://www.jstor.org/stable/1969443 .

Define the function :

$$f(x)=\left(\left(1+x+\frac{1}{x+1}+\frac{1}{1+\frac{1}{x+1}}\right)\right)^{-\frac{1}{2}}$$

Then the infinite composition of $f(x)$ with itself gives the root and $x=1$