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

functional-equations

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.


Solution 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...

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