Is there a closed form expression for the coefficients (after Taylor expansion) of the Beta function $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ around an arbitrary negative integer let's say $x=-k$ and $y=-l$? I want an expression of the coefficients of the expansion in terms of $k$ and $l$.
Is there an analogous expression for $\partial_x\partial_y B(x,y)$ around an arbitrary negative integer $x=-k$ and $y=-l$?