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(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

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

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

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

(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

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

Since from the definition it is clear that $D(a,\vec{b})$ is antisymmetric in the variables $x_j=(N+b_j-j)$, it should be proportional to the Vandermonde of the $x$.

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

This is antisymmetric in both $x_j=(N+b_j-j)$ and $y_i=(N+a_i-i)$, so it should be proportional to both Vandermondes of the $x$ and the $y$...

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

(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

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

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

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

(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

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

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

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

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.

(EDIT: I have removed the denominators I had in a previous version as they were superfluous)

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

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

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