Gamma function manipulation $\Gamma\left(x+\frac{1}{2}\right) = \text{something}\Gamma(x)$?

Question

Is it possible to write $$\Gamma\left(x+\frac{1}{2}\right)$$ in terms of $$\Gamma(x)?$$ I am currently doing an induction argument and I require this, but haven't been able to figure out a nice manipulation and there is nothing useful on Wikipedia.

@CarlSchildkraut I need to be able to represent $\Gamma(x+\frac{1}{2}) = \alpha \Gamma(x)$, where $\alpha$ is a constant or $\Gamma$ function itself. — , Sep 17 '17 at 01:20
@Alanna it's not constant, and I don't believe it can be expressed as $\Gamma(f(x))$ where $f$ is a "nice" function. Can you be specific in what you need it for? — Carl Schildkraut, Sep 17 '17 at 02:04
@reuns Yes, but this evaluates their product, not their sum. If $x$ is an integer this is useful, but it's no more useful to express the quotient in terms of $\Gamma(2x)$ and $\Gamma(x)$ instead of $\Gamma(x+1/2)$ and $\Gamma(x)$. — Carl Schildkraut, Sep 17 '17 at 02:29
@CarlSchildkraut I don't see what you mean at all. $\Gamma(s)$ is a meromorphic function, the duplication/product formula is a deep property more or less equivalent to $\frac{\Gamma'(s)}{\Gamma(s)} = C-\sum_{n=0}^\infty \frac{1}{s+n}-\frac{1}{n}$ — reuns, Sep 17 '17 at 02:32
Concerning the asymptotic expansion $$\frac{\Gamma(x+1/2)}{\Gamma(x)}=\sqrt{x}\left(1-\frac 1{2^3 x}+\frac 1{2^7 x^2}+\frac 5{2^{10} x^3}-\cdots\right)$$ see OEIS A143503 and Dyson, Frankel and Glasser's paper. — Raymond Manzoni, Sep 20 '17 at 07:17

score 3 · Answer 1 · answered Sep 17 '17 at 03:52

If you have a look here, you will find that, for non-negative integer values of $n$, using the reflection formula, we have:$$\Gamma\left(n+\tfrac12\right) = {(2n)! \over 4^n \,n!}\, \sqrt{\pi} = \frac{(2n-1)!!}{2^n} \sqrt{\pi} = {n-\frac12 \choose n}\, n!\,\sqrt{\pi}= \sqrt{\pi}\,{n-\frac12 \choose n} n \,\Gamma(n)$$

Gamma function manipulation $\Gamma\left(x+\frac{1}{2}\right) = \text{something}\Gamma(x)$?

1 Answers1