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 $f$:
$$ \sqrt{f(n)+\sqrt{f(2n)+\sqrt{f(3n)+\sqrt{f(4n)+\cdots}}}}? $$
