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 lambda(N) of the (N+1) x (N+1) Hankel matrix, H-N = (mu(j+k))0 <= j,k <= N, generated by the gamma dependent Jacobi weight w(z, gamma) = e(-gamma z)z alpha(1-z)(beta), z is an element of [0,1], gamma is an element of R, alpha > -1, beta > -1. Applying the arguments of Szego, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P-N(z),z is an element of C\[0,1], with the weight w(z,gamma) = e(-gamma z) z(alpha)(1-z)(beta). Using the polynomials P-N(z), we obtain the theoretical expression of lambda(N), for large N. We also display the smallest eigenvalue lambda(N) for sufficiently large N, computed numerically.