Generalizations of Chebyshev polynomials and polynomial mappings

Authors:
Yang Chen, James Griffin and Mourad E.H. Ismail

Journal:
Trans. Amer. Math. Soc. **359** (2007), 4787-4828

MSC (2000):
Primary 33C45

Published electronically:
May 17, 2007

MathSciNet review:
2320652

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show how polynomial mappings of degree from a union of disjoint intervals onto generate a countable number of special cases of generalizations of Chebyshev polynomials. We also derive a new expression for these generalized Chebyshev polynomials for any genus , from which the coefficients of can be found explicitly in terms of the branch points and the recurrence coefficients. We find that this representation is useful for specializing to polynomial mapping cases for small where we will have explicit expressions for the recurrence coefficients in terms of the branch points. We study in detail certain special cases of the polynomials for small degree mappings and prove a theorem concerning the location of the zeroes of the polynomials. We also derive an explicit expression for the discriminant for the genus 1 case of our Chebyshev polynomials that is valid for any configuration of the branch point.

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Additional Information

**Yang Chen**

Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom

Email:
y.chen@ic.ac.uk

**James Griffin**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32826

Address at time of publication:
Department of Mathematics, American University of Sharjah, P.O. Box 26666, United Arab Emirates

Email:
jgriffin@math.ucf.edu, jgriffin@aus.edu

**Mourad E.H. Ismail**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32826

Email:
ismail@math.ucf.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04022-6

Received by editor(s):
January 27, 2004

Received by editor(s) in revised form:
May 3, 2005

Published electronically:
May 17, 2007

Article copyright:
© Copyright 2007
American Mathematical Society