The $N\times N$ determinant $$D(a,\vec{b})=\det\left( \frac{(2N+a+b_j-i-j)!}{(N-j)!(N+a-i)!}\right)$$ has the nice form $$D(a,\vec{b})=\prod_{j=1}^N\frac{(N+a+b_j-j)!}{(N+a-j)!}\prod_{i=j+1}^N\frac{(b_i+b_j-j+i)}{(i-j)}.$$

I would like to know if the generalization where $a$ is allowed to vary with $i$ has a nice expression as well, $$D(\vec{a},\vec{b})=\det\left( \frac{(2N+a_i+b_j-i-j)!}{(N-j)!(N+a_i-i)!}\right)=?$$

I know Krattenthaler has this great paper about determinants, but I was not able to find help there.