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.