On Stieltjes polynomials

A system of polynomials fE n+1 g which satisfy the or-thogonality condition Z b a P n (x)E n+1 (x) x k h(x) dx = 0; k = 1; : : :; n; where for the weight function h we have h 0, 0 < R b a h(x) dx < 1, with nite moments h n = R b a x n h(x) dx. fP n g is the system of orthogonal polynomials associated with h. The degree of E n+1 is equal to its index n + 1. The orthogonality conditions deene E n+1 up to a multiplicative constant, but the conditions for h given above are not suucient for E n+1 having real zeros which lie in a; b]. However, several special cases and classes of weight functions h are known for which the zeros of the corresponding Stieltjes polynomials do not only have this property, but also interlace with the zeros of P n. A simple example is a; b] = ?1; 1], h(x) = p 1 ? x 2 , P n (x) = U n (x), the Chebyshev polynomial of the second kind, where E n+1 (x) = T n+1 (x), the Chebyshev polynomial of the rst kind. For h(x) = (1 ? x 2) ?1=2 , P n (x) = T n (x) and E n+1 (x) = (1 ? x 2)U n?1 (x). A generalisation are the Bernstein-Szegg o weight functions h(x) = (1 ?x 2) 1=2 == m (x), where m is a polynomial of degree m that is positive in ?1; 1] 6, 7]. Weight functions for which log h= p 1 ? x 2 2 L 1 ?1; 1], p 1 ? x 2 h 2 C?1; 1] and p 1 ? x 2 w(x) > 0 for x 2 ?1; 1] are another class for which the above properties are known to hold asymptotically under certain additional conditions on h 8]. The classical case which was originally considered by Stieltjes is the Legendre weight function h 1, a; b] = ?1; 1]. For this case Szegg o 9] proved that all zeros are real and in the open interval (?1; 1) and that they inter-lace with the zeros of the Legendre polynomials P n. Szegg o extended his proof to the ultraspherical or Gegen-bauer weight function, h(x) = (1 ? x 2) ?1=2 , 2 0; 2], a; …