Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product

Let mu be a finite positive Borel measure supported on [- 1, 1] and introduce the discrete Sobolev-type inner product [f, g] = integral(1)(-1) f(x)g(x)d mu(x) + Sigma(K)(k=1) Sigma(Nk)(i=0) M(k,i)f((i))(a(k))g((i))(a(k)), where the mass points a(k) belong to [-1, 1], and M(k,i) > 0(i = 0, 1,..., N(k)). In this paper, we obtain generalized trace formula and asymptotics of the averaged Turan determinant for the Sobolev-type orthogonal polynomials. Asymptotics of the recurrence coefficients for symmetric Gegenbauer-Sobolev orthogonal polynomials is obtained. Trace formula and asymptotics of Turan's determinant for Gegenbauer-Sobolev orthogonal polynomials are also given. (C) 2006 Elsevier Inc. All rights reserved.