I wonder if there is a closed form for $ \Gamma(a-x)$.
The closed form is known when $a=1$ being the famous formula $\frac{\pi}{\Gamma(x)\sin(\pi x)}$ (and automatically the closed form exist for all integers $a$). Since $\Gamma\left( x+\frac{1}{2}\right)\Gamma\left( x\right)= \frac{\sqrt{\pi} \ \Gamma(2x)}{{2^{2x-1}}}$ one can find the formula when $a=\frac{1}{2}$ being $$\frac{\sqrt{\pi} \ \Gamma(x) 2^{2x-1} }{\sin\left(\pi x +\frac{\pi}{2}\right)\Gamma(2x)}$$(also the closed form exist for all $n+ \frac{1}{2}$ where $n \in \mathbb{N}$ in general we need to find a closed form for $a\in(0,1]$)
I don't think a closed form exist for all $a$ but at least they exist for some values. Is there is any other closed form for other values than $1, \frac 1 2 $? I couldn't find closed form for any other values.
And by closed form here I mean a finite combinations of elementary functions, powers of $\Gamma(a)$ and powers of $\Gamma(kx)$ where $k$ is real, any other representation involving any other special functions like beta are not what I am looking for.