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Let $x$ be a real number and $f(x)$ a real analytic function such that $f(x+1) = P(f(x),x)$ where $P$ is a given real polynomial. Express $f(x)$ as an integral from $0$ to $\infty$.

As an example we have the $\Gamma$ function. http://mathworld.wolfram.com/GammaFunction.html

I'm looking for the general solution.

I was thinking about the recursions used to compute integrals of type $\int$ $f(x)^k \mathrm{d}x$.

Also hypergeometric functions crossed my mind.

But I am stuck.

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    If we replace/approximate $f(x+1)$ with a truncated Taylor series of $f(x+1)$ in terms of $f(x)$, we get a differential equation. Not sure if that works well though.2013-06-27

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