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

Let μμ be a finite positive Borel measure supported on [-1,1][-1,1] and introduce the discrete Sobolev-type inner product〈f,g〉=∫-11f(x)g(x)dμ(x)+∑k=1K∑i=0NkMk,if(i)(ak)g(i)(ak),where the mass points akak belong to [-1,1][-1,1], and Mk,i>0(i=0,1,…,Nk)Mk,i>0(i=0,1,…,Nk). 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.