Jacobi polynomials as generalized Faber polynomials

Author:
Ahmed I. Zayed

Journal:
Trans. Amer. Math. Soc. **321** (1990), 363-378

MSC:
Primary 33A65; Secondary 30C20

MathSciNet review:
965745

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an open bounded subset of the complex -plane with closure whose complement is a simply connected domain on the Riemann sphere. map the domain one-to-one conformally onto the domain such that . Let , be analytic in the domain with . Let , be analytic in and analytically continuable to any point outside along any path not passing through the points .

The generalized Faber polynomials of are defined by

(1) if the Jacobi polynomials are generalized Faber polynomials of any region , then it must be the elliptic region

(2) the only Jacobi polynomials that can be classified as generalized Faber polynomials are the Tchebycheff polynomials of the first kind, some normalized Gegenbauer polynomials, some normalized Jacobi polynomials of type , and there are no others, no matter how one normalizes them;

(3) the Hermite and Laguerre polynomials cannot be generalized Faber polynomials of any region.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1990-0965745-1

Keywords:
Jacobi polynomials,
Faber polynomials

Article copyright:
© Copyright 1990
American Mathematical Society