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Perturbed Hankel Determinants

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_{α,β}h] is compared with the known determinant D_n[w_{α,β}]. Here w_{α,β} is the Jacobi weight and w_{α,β}h, where h=h(x),x\in[-1,1] is strictly positive and real analytic, is the smooth perturbation on the Jacobi weight w_{α,β}(x):=(1-x)^α(1+x)^β. Applying a previously known formula on the distribution function of linear statistics, we compute the large n asymptotics of D_n[w_{α,β}h] and supply a missing constant of the expansion.