TITLE: Variables inside nested radicals
QUESTION [8 upvotes]: It is well known that nested radicals that contain constant values can be "de-nested" using known formulae or have alternate ways of being expressed. For instance, we have:
$$
{\sqrt  {n+{\sqrt  {n+{\sqrt  {n+{\sqrt  {n+\cdots }}}}}}}}={\tfrac  12}\left(1+{\sqrt  {1+4n}}\right), \\
 \sqrt{n-\sqrt{n-\sqrt{n-\sqrt{n-\cdots}}}} = \tfrac12\left(-1 + \sqrt {1+4n}\right), \\
\ldots
$$
There are also some general formulae such as:
$$
\sqrt{a-b\sqrt{a-b\sqrt{a-\cdots}}}=\cfrac{a}{b+\cfrac{a}{b+\cfrac{a}{b+\cdots}}},
$$
which is valid for $a>b>0$.
However, I could not find any expressions or formulae involving variable quantities within the radicals. Are there any known formulae for expressions like:
$$
\sqrt{n+\sqrt{n^2+\sqrt{n^3+\sqrt{n^4+\cdots}}}}?
$$
Or nested radicals with certain values of a given function $f$:
$$
\sqrt{f(n)+\sqrt{f(2n)+\sqrt{f(3n)+\sqrt{f(4n)+\cdots}}}}?
$$

REPLY [2 votes]: I find http://www.dgp.toronto.edu/~mjmcguff/math/nestedRadicals.pdf an excelent treatment of the question.