On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials

Abstract

With the aid of Havin’s Lemma (which we generalize) we prove that polynomials orthogonal over the unit disk with respect to certain weighted area measures (Bergman polynomials) cannot satisfy a finite-term recurrence relation unless the weight is radial, in which case the polynomials are simply monomials. For polynomials orthogonal over the unit circle (Szegő polynomials) we provide a simple argument to show that the existence of a finite-term recurrence implies that the weight must be the reciprocal of the square modulus of a polynomial.

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Correspondence to Laurent Baratchart.

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Dedicated to Nicolas Papamichael, a great mentor, collaborator, and friend.

The work of the second author was supported, in part, from the U.S. National Science Foundation grant DMS-0808093.

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Baratchart, L., Saff, E.B. & Stylianopoulos, N.S. On Finite-Term Recurrence Relations for Bergman and Szegő Polynomials. Comput. Methods Funct. Theory 12, 393–402 (2012). https://doi.org/10.1007/BF03321834

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En]Keywords

  • Bergman orthogonal polynomials
  • recurrence relations
  • weights

2000 MSC

  • 30C10
  • 30C30
  • 30C50
  • 30C62
  • 41A10