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Your quantity is $$ P(n,\alpha)=\det_r A_{i,j}(n,\alpha),$$ with $$ A_{i,j}(n,\alpha)=\int_0^\alpha t^{n+r-i-j}e^{-t}dt.$$ By the Andreief identity, this is $$ P(n,\alpha)=\frac{1}{r!}\int_0^\alpha e^ …

The Selberg integral, $$S_{n} (\alpha, \beta, \gamma) = \int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n$$ in …

I have found a solution myself, at least in the case when $a$ and $b$ are partitions. The determinant can be written as $$ D=\det((x_i+y_j)!)=\det\left( \int z^{x_i+y_j}e^{-z}dz\right)$$ We resort t …

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

The case $k=n$ is a consequence of the identity $$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$$ which I have seen under the names "And …