The smallest eigenvalue of large Hankel matrices.

We investigate the large N behavior of the smallest eigenvalue, λ N , of an ( N + 1 ) × ( N + 1 ) Hankel (or moments) matrix H N , generated by the weight w ( x ) = x α ( 1 − x ) β , x ∈ [ 0 , 1 ] , α > − 1 , β > − 1 . By applying the arguments of Szegö, Widom and Wilf, we establish the asymptotic formula for the orthonormal polynomials P n ( z ) , z ∈ C ∖ [ 0 , 1 ] , associated with w ( x ) , which are required in the determination of λ N . Based on this formula, we produce the expressions for λ N , for large N . Using the parallel algorithm presented by Emmart, Chen and Weems, we show that the theoretical results are in close proximity to the numerical results for sufficiently large N . [ABSTRACT FROM AUTHOR]