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

Full-text PDF Free Access

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]**R. Askey,*Orthogonal polynomials and special functions*, Regional Conference Series in Appl. Math., vol. 21, SIAM, 1975. MR**0481145 (58:1288)****[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. Gilbert,*Integral operator methods in bi-axially symmetric potential theory*, Contrib. Differential Equations**2**(1963), 441-456. MR**0156998 (28:239)****[5]**-,*Bergman's integral operator method in generalized axially symmetric potential theory*, J. Math. Phys.**5**(1964), 983-997. MR**0165131 (29:2420)****[6]**Z. Nehari,*On the singularities of Legendre expansions*, Indiana Math. J.**5**(1956), 987-992. MR**0080747 (18:293d)****[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. Suetin,*The basic properties of Faber polynomials*, Uspekhi Mat. Nauk**19**(1964), No.4, 125-154; Russian Math. Surveys**19**(1964), No.4, 121-149. MR**0168773 (29:6029)****[10]**-,*Fundamental properties of polynomials orthogonal on a contour*, Usepkhi Mat. Nauk**21**(1966), No.2 (128), 41-88; Russian Math. Surveys**21**(1966), No.2, 35-83. MR**0198111 (33:6270)****[11]**-,*Polynomials orthogonal over a region and Bieberbach polynomials*, Proc. Steklov Inst. Math., No. 100 (1971); English transl., Amer. Math. Soc., Providence, R. I., 1974. MR**0463793 (57:3732b)****[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 revisted*, Complex Variables Theory Appl.**10**(1988), 11-22. MR**946095 (89g:30006)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
33A65,
30C20

Retrieve articles in all journals with MSC: 33A65, 30C20

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