Blumenthal's theorem for Laurent orgthogonal polynomials

  • Authors:

  • A. Sri Ranga;Walter Van Assche

  • Affiliations:

  • DCCE-IBILCE, Universidade estadual Paulista, Rua Cristóvão Colombo, São José do Rio Preto, SP, Brazil;Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan, Leuven, Belgium

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2002

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Abstract

We investigate polynomials satisfying a three-term recurrence relation of the form Bn(x) = (x-βn)Bn-1(x)-αnxBn-2(x), with positive recurrence coefficients αn+1,βn (n = 1,2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where αn→α and βn→β show that the zeros of Bn, are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.