My friend came to see me regarding some calculation in probability where he would like to know if it is possible to solve for a variable analytically under the gamma function. By this I mean say we are given the value of some quantity $x$, such that
$x = \dfrac{\Gamma(1 + \frac{2}{k})}{\left(\Gamma(1 + \frac{1}{k})\right)^2}$
I would like to solve this for some real number $k$. I can use the factorial to manipulate this expression and get
$x = \dfrac{(\frac{2}{k})!}{{\left(\left(\frac{1}{k}\right)!\right)}^2 }$
If (a big if) I can make the substitution $n = \frac{1}{k}$, this would be equal to the central binomial coefficient. However I don't think this is possible as $\frac{1}{k}$ is not an integer in general.
What else can I do? Thanks.