Forget the $p$'s and consider a product $\Gamma(x-\epsilon)\Gamma(y-\epsilon)\ldots\Gamma(z-\epsilon)$. If one of $x,y,\ldots,z$ is not a negative integer, it is regular, and you can put $\epsilon=0$ in the factor. So it is enough to consider the case where all of $x,y,\ldots,z$ are negative integers. In this case, if there are $k$ factors, $\epsilon^k\Gamma(x-\epsilon)\Gamma(y-\epsilon)\ldots\Gamma(z-\epsilon)$ is analytic at $\epsilon=0$.

Therefore you can expand it into a power series using the power series for $\epsilon\Gamma(x-\epsilon)$ etc. to order $k$ or higher. At the end substitute for $x,y,\ldots,z$ your expressions in $p$,