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][1] about determinants, but I was not able to find help there.


  [1]: http://emis.ams.org/journals/SLC/wpapers/s42kratt.pdf