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Fractional iterates of x²-2, a problem by Ramanujan

A curious result, attributed to Ramanujan, is stated here. The statement is: if \(F(x)=x^2-2\), where \(x>2\) then its fractional iterates are: $$F^{[\log_23]}(x) = x^3-3x,$$ $$F^{[\log_25]}(x) = x^5 - 5x^3+5x.$$ Amazing result, indeed. Unfortunately, the blog post tells nothing about the way these solutions obtained, and I have no easy way to access Ramanujan's notebooks. The topic of fractional iterations of functions always fascinated me, so I decided to try my hand in solving it. This post is not about the final solution (which turned out to be surprisingly simple), but about the way I solved it and enjoyed the solution. In the course I rediscovered some fact that, retrospectively, can be seen from the very beginning. What are fractional iterates by the wayLet \(F(x)\) be some function. Then its second iterate is the function $$F^{[2]}(x) = F(F(x)).$$ (In this post to denote iteration I use upper index in square…

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