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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.

gebruiker
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Elmehdi
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    see https://en.wikipedia.org/wiki/Gamma_function#Properties. you can't if $r$ is not an integer, and if it is rational $r = a + b/m$, you have to express it in term of $\Gamma(m(x+a))$ and of $\Gamma(x+a+k/m)$ for $k \in {0 \ldots m-1} \setminus { b}$. – reuns Mar 15 '16 at 12:19

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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

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

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 $\ln(n+r+m-1)+\ln(n+r+m-2)+\cdots+\ln(n+r).$