How to evaluate $\frac{d}{dx}{x!}$

Question

How to evaluate that? I have seen some solution using $\gamma$, but i don’t really understand that.

Answer 1

The factorial of a number could be extended to real numbers using Gamma function :
$\Gamma(x+1) = x! = \displaystyle\int_0^{\infty} t^x e^{-t} dt$

So we could find the derivative of the factorial of a real number using the intergral representation of the Gamma function:

\begin{aligned}\dfrac{d}{dx} x!&=\dfrac{\partial}{\partial x}\displaystyle\int_0^{\infty} t^x e^{-t} dt\\&=\displaystyle\int_0^{\infty} \Big(\dfrac{\partial }{\partial x}t^x\Big) e^{-t} dt\\&=\displaystyle\int_0^{\infty} t^x \ln t e^{-t} dt\end{aligned}

There is no a closed form for the latter expression,however, it could be expressed in terms of the Gamma function and the Digamma function $\psi(x)$ :

$\dfrac{d}{dx} x! = \Gamma'(x+1) = \Gamma(x+1) \psi(x+1)$

The integral representation of the digamma function is given by:

${\displaystyle \psi (x)=\int _{0}^{\infty }\left({\frac {e^{-t}}{t}}-{\frac {e^{-xt}}{1-e^{-t}}}\right)\,dt.}$