Does the incomplete beta function $B(z,a,b)$ still hold even for negative values of $b$? For instance, consider
$$\rho(r)=\frac{b_{0}}{1-q}B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$$
where $B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$ is an incomplete beta function, $b_{0}$ is some positive constant, $-\infty<q<1$. For my range of $q$, the parameter $\frac{1}{q-1}$ is always negative.
Based from http://functions.wolfram.com/GammaBetaErf/Beta3/02/01/ , $b>0$.