how to rewrite $\Gamma(x+r)$ in terms of $\Gamma(x)$

Question

I'm looking if there is a transformation of the gamma function applied to $x+r$ $\Gamma(x+r)$ in terms of $\Gamma(x)$, where $r$ is a positive real number.

Answer 1

There is the well-known recurrence formula when $r=1$ $$\Gamma(x+1)=x\Gamma(x),$$ which you can extend to $\Gamma(x+n)$.

For noninteger $r$, there is no simple relation. Just notice the duplication formula

$$\Gamma(x+\frac12)=2^{1-2x}\sqrt\pi\frac{\Gamma(2x)}{\Gamma(x)}.$$

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You can get a fairly good approximation of $\ln(\Gamma(n+r))$ where $\lfloor r\rfloor=0$ by linearly interpolating between $\ln(\Gamma(n+m))$ and $\ln(\Gamma(n+1+m))=\ln(n+m)+\ln(\Gamma(n+m))$ for some integer $m$, and deducing $\ln(n+r+m-1)+\ln(n+r+m-2)+\cdots+\ln(n+r).$