Jacobi polynomials as generalized Faber polynomials

Author:
Ahmed I. Zayed

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

MSC:
Primary 33A65; Secondary 30C20

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

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.

**[1]**Richard Askey,*Orthogonal polynomials and special functions*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0481145****[2]**G. Faber,*Über polynomische Entwicklungen*, Math. Ann.**57**(1903), 389-408;**64**(1907), 116-135.**[3]**Ya. L. Geronimus,*Polynomials, orthogonal on a a circle and on an interval*, Fizmatgiz, Moscow, 1958.**[4]**R. P. Gilbert,*Integral operator methods in bi-axially symmetric potential theory*, Contributions to Differential Equations**2**(1963), 441–456 (1963). MR**0156998****[5]**R. P. Gilbert,*Bergman’s integral operator method in generalized axially symmetric potential theory*, J. Mathematical Phys.**5**(1964), 983–997. MR**0165131**, https://doi.org/10.1063/1.1704199**[6]**Zeev Nehari,*On the singularities of Legendre expansions*, J. Rational Mech. Anal.**5**(1956), 987–992. MR**0080747****[7]**V. J. Smirnov and N. Lebedev,*Functions of a complex variable*, M. I. T. Press, Cambridge, Mass., 1968.**[8]**H. Srivatstave and H. Manocha,*A treatise on generating functions*, Ellis Horwood, West Sussex, England, 1984.**[9]**P. K. Suetin,*The basic porperties of Faber polynomials*, Uspehi Mat. Nauk**19**(1964), no. 4 (118), 125–154 (Russian). MR**0168773****[10]**P. K. Suetin,*Fundamental properties of polynomials orthogonal on a contour*, Uspehi Mat. Nauk**21**(1966), no. 2 (128), 41–88 (Russian). MR**0198111****[11]**P. K. Suetin,*Polynomials orthogonal over a region and Bieberbach polynomials*, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by R. P. Boas. MR**0463793****[12]**G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.**[13]**A. Zayed, M. Freund, and E. Görlich,*A theorem of Nehari revisited*, Complex Variables Theory Appl.**10**(1988), no. 1, 11–22. MR**946095**, https://doi.org/10.1080/17476938808814283

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Additional Information

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

Keywords:
Jacobi polynomials,
Faber polynomials

Article copyright:
© Copyright 1990
American Mathematical Society