Solve $f^2(x)=x+f(x+1)$

Question

If the function $f(x)$ is such that $$f^2(x)=x+f(x+1),$$ find a closed-form expression for $f$.

I found $$f(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2+\sqrt{x+3+\cdots}}}}$$ is such an $f$. Does anyone have other solutions? Thank you.

Answer 1

You can always take either square-root, so $$ f(x)=\pm\sqrt{x\pm\sqrt{x+1\pm\sqrt{x+2\pm\sqrt{x+3\pm\cdots}}}} $$ Gives you uncountably many solutions...

Answer 2

Notice that $f(0)$ is nothing else than the square root of the Nested Radical Constant, which is yet unknown to possess a closed form. Obviously, if f were to possess such a form, then so would this constant, meaning your question is still open.