Closed form for the first local min $\gt 0$ of $x!$ (in reference to x-value of min)

Question

The first local min of $x!$ is the point $(0.461632...,0.885603...)$

Is there a close form of $0.461632...$, the $x$-value of the above point? If you can tell me the closed form, could you help me prove it?

Answer 1

I do not think that there is a closed form for this minimum. If $$f(x)=x!=\Gamma (x+1)$$ $$f'(x)=\Gamma (x+1) \psi ^{(0)}(x+1) $$ and then the minimum happens at the zero of $\psi ^{(0)}(x+1)$ which corresponds, as you wrote to $\approx 0.4616321450$.

Looking at RIES, avery good approximation of this number seems to be $$e^{-e} (3 \phi -1+\pi )\approx 0.4616321473$$

I suggest you have a look at sequence $A030169$ at OEIS.

You could find the continued fraction $(A030170)$ at OEIS.