Asymptotics of polynomials orthogonal with respect to a discrete-complex Sobolev inner product

Let mu be a finite positive Borel measure supported on a compact set of the real line and introduce the discrete Sobolev-type inner product < f,g > = integral f(x) g(x) d mu(x) + (K)Sigma(k=1) (Nk)Sigma(i=o) M-k,M-i f((i)) (c(k)) g(i) (c(k)), where the mass points c(k) belong to supp(mu) and M-k,M-i are complex numbers such that M-k,(Nk) not equal 0. In this paper we investigate the asymptotics of the polynomials orthogonal with this product. When the mass points c(k) belong to C\supp(mu), the problem was solved in a paper by G. Lopez, et al. (Constr. Approx. 11 (1995) 107-137) and, for mass points in supp(mu) = [-1,1], the solution was given by I.A. Rocha et al. (J. Approx. Theory, 121 (2003) 336-356) provided that mu '(x) > 0 a.e. x is an element of [- 1, 1] and M-k,M-i are nonnegative constants. If mu is an element of M(0, 1), the possibility c(k) is an element of supp (mu)\[-1,1] must be considered. Here we solve this last case with complex constants M-k,M-i. (c) 2004 Elsevier B.V. All rights reserved.