Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.

Question

Now asked on MO here.

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I wonder if there is a closed form for $ \Gamma(a-x)$.

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.

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 integer $a$). 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.

Answer 1

I think the following identity is different from the reflection formula and is valid except for arguments for which the $\Gamma$ function is undefined namely the negative integers: $$ \Gamma(a-x) =\frac{\sqrt{\pi}\; \Gamma(2a-2x-1)}{4^{a-x-1}\;\Gamma(a-x-\frac{1}{2})} $$