Does Gamma function a solution for known Ordinary differential equation?

Question

It is well known that gamma function's defined as : $$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$ and it is divergent for $x<0$. , Really I ask about differential equation which Gamma function satisfying it or by Other way : Does Gamma function a solution for known Ordinary differential equation and if yes what is it ? For example if it obeyed any form of $F( \Gamma, \Gamma ', \dots, \Gamma^{(k)}) = 0$ ?

Answer 1

This is a well known result: Hölder's theorem

Answer 2

The gamma function does not satisfy any algebraic differential equation . But it is the solution of the following nonalgebraic differential equation: $$\frac{\partial w(x)}{\partial x}=w(x)~\psi(x);\qquad w(x)=\Gamma(x)$$

Otto Hölder proved in $1887$ that,

The gamma function does not satisfy any algebraic differential equation

by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it a transcendentally transcendental function. This result is known as Hölder's theorem.