fractional iteration, Ramanujan

Ramanujan stated (quaterly reports, Ramanujan’s Notebooks vol. 1) that if F(x) = x^{2} -2, x>2 then the fractional iterates of F are:

F^{1/2}(x) = \left( \frac{x + \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}} - \left( \frac{x - \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}}

F^{\log(3)/\log(2)}(x) = x^{3} - 3x

F^{\log(5)/\log(2)}(x) = x^{5} - 5x^{3} +5x

what’s kind of amazing about this is that some of F‘s transcendental order compositions are polynomials with rational coefficients!

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