Perturbed Hankel determinants

Abstract

In this short note, we compute, for large n the determinant of a class of n x n Hankel matrices, which arise from a smooth perturbation of the Jacobi weight. For this purpose, we employ the same idea used in previous papers, where the unknown determinant, D_n[w_{\alpha,\beta}h] is compared with the known determinant D_n[w_{\alpha,\beta}]. Here w_{\alpha,\beta} is the Jacobi weight and w_{\alpha,\beta}h, where h=h(x),x\in[-1,1] is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight w_{\alpha,\beta}(x):=(1-x)^\alpha (1+x)^\beta. Applying a previously known formula on the distribution function of linear statistics, we compute the large n asymptotics of D_n[w_{\alpha,\beta}h] and supply a missing constant of the expansion.