ASYMPTOTIC BEHAVIOUR OF THE JACOBI SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS. A NON DIAGONAL CASE

Abstract

Asymptotic behaviour of the Jacobi Sobolev-type orthogonal polynomials. A non-diagonal case. Rev. Acad. Colomb. Cienc. 34 (133): 529-539, 2010. ISSN 0370-3908. MATEMÁTICAS 1 Universidad Nacional de Colombia, Bogotá, Colombia. Correo electrónico: haduenasr@unal.edu.co 2 Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo No. 340, Colima, Colima, México. Correo electrónico: garzaleg@gmail.com AMS Classification 2000: 33C47. Consider the following Sobolev type inner product (p, q) = ∫ p'(x)q(x)(1 - x)θ(1 + x)θdx + i[p'(1)q(1) - p(1)q'(1)], (1) where p and q are polynomials with real coefficients, 0 < θ > -1, ξ(x) = (p(x), q(x))', and A = [M0, M1; M1, M2] is a positive semidefinite matrix, with M0M1 ≥ 0, and A ∈ ℝ. The family of polynomials orthogonal with respect to (1), P_n^{(θ,θ)}, are called Jacobi Sobolev-type orthogonal polynomials. An expression that relates this family of polynomials with P_n^{(0,0)}, the usual Jacobi orthogonal polynomials, was obtained in [8]. Here, we obtain the outer relative asymptotic for P_n^{(0,0)}, as well as the corresponding Mehlér-Heine formula.