## Generalizations of Chebyshev polynomials and polynomial mappings

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- by Yang Chen, James Griffin and Mourad E.H. Ismail PDF
- Trans. Amer. Math. Soc.
**359**(2007), 4787-4828 Request permission

## 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

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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
- MR Author ID: 91855
- Email: ismail@math.ucf.edu
- Received by editor(s): January 27, 2004
- Received by editor(s) in revised form: May 3, 2005
- Published electronically: May 17, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**359**(2007), 4787-4828 - MSC (2000): Primary 33C45
- DOI: https://doi.org/10.1090/S0002-9947-07-04022-6
- MathSciNet review: 2320652