Szegő-type determinant asymptotics for generalized Hilbert matrices with edge eigenvalues

We compute the large-$N$ asymptotics of the determinant of the identity plus or minus a generalized Hilbert matrix. For $N \in \mathbb{N}$ and $δ\in \mathbb{R} \setminus \{-1, -2, \ldots\}$, let \[ H_N^δ := \left( \frac{\sin((1+δ)π)}{π(j+k+1+δ)} \right)_{j,k=0}^{N-1} \] be the generalized