Determinants of Hankel matrices (Q5927512)

The problem how to compute asymptotically the determinants of the finite matrices \(H_n(u)\) defined by \(\text{det} (a_{i+j})_{i,j=0}^{n-1}\), where NEWLINE\[NEWLINEa_{i+j}=\int _0^{\infty}x^{i+j}u(x) dx NEWLINE\]NEWLINE and \(u(x)\) is a suitable weight function on a semi-definite interval, is considered. The entries of these determinants depend only on the sum \(i+j\) and are classified as Hankel determinants. The problem of calculating Hankel determinants is solved using asymptotics for certain known integral operators i. e. the kernel of the integral operator is approximated by a different kernel which involves Bessel functions.