Is there a rigorous proof for the analytic continuation of the incomplete beta function $B(x;a,b)$ for all values of $a$ and $b$? The incomplete beta function normally restricts the values of $a,b$ as $a>0$, $b>0$. So I would like to extend these values to negative but I cannot find a good and comprehensive reference for this.
There is a given expression in wikipedia for the analytic continuation of beta function but I cannot see one for the incomplete one. By the way, the integral that I am working on is:
And I am working on $-\infty<q<1$. Thanks for the help.