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

Let μ 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 〉 = ∫ f ( x ) g ( x ) d μ ( x ) + ∑ k = 1 K ∑ i = 0 N k M k , i f ( i ) ( c k ) g ( i ) ( c k ) , where the mass points c k belong to supp ( μ ) and M k , i are complex numbers such that M k , N k ≠ 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 ( μ ) , the problem was solved in a paper by G. López, et al. (Constr. Approx. 11 (1995) 107–137) and, for mass points in supp ( μ ) = [ - 1 , 1 ] , the solution was given by I.A. Rocha et al. (J. Approx. Theory, 121 (2003) 336–356) provided that μ ′ ( x ) > 0 a.e. x ∈ [ - 1 , 1 ] and M k , i are nonnegative constants. If μ ∈ M ( 0 , 1 ) , the possibility c k ∈ supp ( μ ) ⧹ [ - 1 , 1 ] must be considered. Here we solve this last case with complex constants M k , i .