Infinitely nested radical expansion of functions

Is where a 'best' way to make a nested radical expansion for an analytic function? This way seems convenient:

$$f(x)=a_0+a_1x+a_2x^2+a_3x^3+\dots=\sqrt{a_0^2+2a_0a_1x+(a_1^2+2a_0a_2)x^2+\cdots}=$$

$$=\sqrt{a_0^2+2a_0a_1x+(a_1^2+2a_0a_2)x^2 \sqrt{1+\cdots}}$$

I conjecture that this infinitely nested radical expansion has the same interval of convergence as the original Taylor series. Is this correct?

Also, the number of 'roots' we take into account gives us twice the number plus one of the correct terms in the Taylor series.

For example:

$$e^x=\sqrt{1+2x+2x^2\sqrt{1+\frac{4}{3}x+\frac{10}{9}x^2\sqrt{1+\frac{32}{25}x+\frac{681}{625}x^2\sqrt{1+\dots}}}}=$$

$$=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\dots$$

This expression converges for any $x$.

$$\frac{1}{1-x}=\sqrt{1+2x+3x^2\sqrt{1+\frac{8}{3}x+\frac{46}{9}x^2\sqrt{1+\frac{76}{23}x+\frac{4089}{529}x^2\sqrt{1+\dots}}}}=$$

$$=1+x+x^2+x^3+x^4+x^5+x^6+\dots$$

This expression converges for $|x|<1$.

This idea may seem pointless, since the coefficients in the radical expansion are very hard to calculate even for functions with 'simple' Taylor series.

But is it possible, that some function with 'complicated' Taylor series will have simple nested radical expansion of this kind?

For example, consider the easiest nested radical of this kind:

$$f(x)=\sqrt{1+x+x^2\sqrt{1+x+x^2\sqrt{1+x+x^2\sqrt{1+\dots}}}}=$$

$$=1+\frac{x}{2}+\frac{3x^2}{8}+\frac{x^3}{16}+\frac{11x^4}{128}-\frac{9x^5}{256}+\frac{27x^6}{1024}+\dots+$$

Though this kind of functions we can always find in closed form, if we assume the infinite nested radical converges.

$$f(x)=\sqrt{1+a_1x+a_2x^2\sqrt{1+a_1x+a_2x^2\sqrt{1+a_1x+a_2x^2\sqrt{1+\dots}}}}=$$

$$=\frac{a_2}{2}x^2+\sqrt{1+a_1x+\frac{a_2^2}{4}x^4}$$

I've never seen this topic discussed anywhere, so a reference would be nice. The only thing I've seen is Ramanujan nested radical, and it's usually presented as a funny trick, nothing more.

This is a really interesting question!

I can't find any textbooks, but the following links to papers look helpful:

http://www.fq.math.ca/Papers1/45-3/osler.pdf

It seems like it is an area with room for much more research.

Also see the references from this webpage: http://mathworld.wolfram.com/NestedRadical.html

Finally, maybe it is possible to establish some conclusions using the theory of continued fractions, say by a logarithm-like transformation?

https://en.wikipedia.org/wiki/Continued_fraction

1. Nested Square Roots of 2 Author(s): L. D. Servi Source: The American Mathematical Monthly, Vol. 110, No. 4 (Apr., 2003), pp. 326-330 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/3647881 .

2. Journal of Approximation Theory 174 (2013) 90–112 On Viete-like formulas ￼￼￼￼￼￼￼￼￼￼￼￼Samuel G. Moreno∗, Esther M. Garcıa-Caballero Departamento de Matematicas, Universidad de Jaen, 23071 Jaen, Spain Received 7 November 2012; received in revised form 6 June 2013; accepted 27 June 2013 Available online 11 July 2013

3. Continued Radicals Author(s): Edward J. Allen Source: The Mathematical Gazette, Vol. 69, No. 450 (Dec., 1985), pp. 261-263 Published by: Mathematical Association Stable URL: http://www.jstor.org/stable/3617569

4. THE RAMANUJAN JOURNAL, 10, 305–324, 2005 Springer Science + Business Media, Inc. Manufactured in the Netherlands.

A New Class of Infinite Products Generalizing Viete’s Product Formula for π AARON LEVIN adlevin@math.brown.edu

Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912 Received November 11, 2002; Accepted May 21, 2004

Interesting related post: Can the general septic be solved by infinitely nested radicals?

• Thank you for the links, but I'm wary of file sharing sites (especially the ones that ask me to enter my e-mail) - could you please just provide the references for the papers - I will then download them myself – Yuriy S Apr 11 '16 at 6:52
• message me if you're having trouble finding them/getting access – Chill2Macht Apr 11 '16 at 13:40