Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show how polynomial mappings of degree $ \mathfrak{K}$ from a union of disjoint intervals onto $ [-1,1]$ 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 $ g$, from which the coefficients of $ x^n$ 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 $ \mathfrak{K}$ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 33C45

Retrieve articles in all journals with MSC (2000): 33C45


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: http://dx.doi.org/10.1090/S0002-9947-07-04022-6
PII: S 0002-9947(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