Some complete elliptic integral of first and second kind $E(k)$ and $K(k)$ can be evaluated for some particular values of $k$ in terms of Euler Gamma function. For example, for $k = \sqrt{2}/2$, $E(k)$ and $K(k)$ can be evaluated in terms of $\Gamma(1/4)$. Regarding this topic, I raise the following two questions:
1 - I am curious, about the mathematical procedure which is implemented to transform the elliptic integral to an integral solvable using the Gamma function. I was able to figure out the transformation for the case $k = \sqrt{2}/2$, but couldn't do it for other values of $k$ for which I know that their corresponding elliptic integrals are indeed expressed in terms of Gamma function (e.g. $k = \frac{\sqrt{6} - \sqrt{2}}{4}$). What is the sequence of transformations applied to the elliptic integral to generate the resulting integral solvable in terms of Gamma function (Hint: The Euler Beta function $B(x,y)$ is indeed involved in this evaluation)?
2 - Is there a mathematical formula through which we can tell the values of $k$ whose corresponding elliptic integrals $K(k)$ and $E(k)$ are solvable by Euler Gamma function? and if so, is there a direct solution to the integral in terms of $k$?
Please support your answers with necessary references whenever possible.
Thanks in advance for your help.