The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

Abstract

In this paper, we study the large N behavior of the smallest eigenvalue λN of the (N+1)×(N+1) Hankel matrix, HN=(μj+k)0≤j,k≤N, generated by the γ dependent Jacobi weight w(z,γ)=e−γzzα(1−z)β,z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials PN(z), z∈C﹨[0,1], with the weight w(z,γ)=e−γzzα(1−z)β. Using the polynomials PN(z), we obtain the theoretical expression of λN, for large N. We also display the smallest eigenvalue λN for sufficiently large N, computed numerically.