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

In this paper, we study the large N behavior of the smallest eigenvalue λ N of the (N + 1) × (N + 1) Hankel matrix, H N = (μ j + k) 0 ≤ j , k ≤ N , generated by the γ dependent Jacobi weight w (z , γ) = e − γ z z α (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 P N (z) , z ∈ C ﹨ [ 0 , 1 ] , with the weight w (z , γ) = e − γ z z α (1 − z) β. Using the polynomials P N (z) , we obtain the theoretical expression of λ N , for large N. We also display the smallest eigenvalue λ N for sufficiently large N , computed numerically. [ABSTRACT FROM AUTHOR]