On determinants identity minus Hankel matrix

In this note, we study the asymptotics of the determinant $\det(I_N - \beta H_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq 1$. Moreover, we assume $\beta\in\mathbb C$ with $|\beta|<1$ and $I_N$ denotes the identity matrix. We determine the first order asymtoptics as $N\to\infty$ of such determinants and show that they exhibit power-like asymptotic behaviour, with exponent depending on the height of the jumps. For example, for the $N \times N$ truncation of the Hilbert matrix $\mathbf{H}$ with matrix elements $\pi^{-1}(j+k+1)^{-1}$, where $j,k\in \mathbb Z_+$ we obtain $$\log \det(I_N - \beta \mathbf{H}_N) = -\frac{\log N}{2\pi^2} \big(\pi\arcsin(\beta)+\arcsin^2(\beta)+o(1)\big),\qquad N\to\infty.$$