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

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

Published electronically:
May 17, 2007

MathSciNet review:
2320652

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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.

**1.**N. I. Ahiezer,*Orthogonal polynomials on several intervals*, Soviet Math. Dokl.**1**(1960), 989–992. MR**0110916****2.**N. I. Akhiezer,*The classical moment problem and some related questions in analysis*, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR**0184042****3.**N. I. Ahiezer and Ju. Ja. Tomčuk,*On the theory of orthogonal polynomials over several intervals*, Dokl. Akad. Nauk SSSR**138**(1961), 743–746 (Russian). MR**0131005****4.**N. I. Akhiezer,*Elements of the theory of elliptic functions*, Translations of Mathematical Monographs, vol. 79, American Mathematical Society, Providence, RI, 1990. Translated from the second Russian edition by H. H. McFaden. MR**1054205****5.**W. Al-Salam, W. Allaway, and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc.**284**(1984), 39-55.MR**0742411 (85j:33005)****6.**J. A. Charris and M. E. H. Ismail, On sieved Orthogonal Polynomials II : Randon walk polynomials, Canad. J. Math.**38**(1986) 397-415. MR**0833576 (87j:33014a)****7.**Jairo A. Charris, Mourad E. H. Ismail, and Sergio Monsalve,*On sieved orthogonal polynomials. X. General blocks of recurrence relations*, Pacific J. Math.**163**(1994), no. 2, 237–267. MR**1262296****8.**Yang Chen and Mourad E. H. Ismail,*Ladder operators and differential equations for orthogonal polynomials*, J. Phys. A**30**(1997), no. 22, 7817–7829. MR**1616931**, https://doi.org/10.1088/0305-4470/30/22/020**9.**Yang Chen and Nigel Lawrence,*A generalization of the Chebyshev polynomials*, J. Phys. A**35**(2002), no. 22, 4651–4699. MR**1908638**, https://doi.org/10.1088/0305-4470/35/22/302**10.**J.S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc.**308**(1988) 559-581. MR**0951620 (89f:42021)****11.**Ya. L. Geronimus, On some finite difference equations and corresponding systems of orthogonal polynomials, Mem. Math. Sect. Fac. Phys. Kharkov State Univ. Kharkov Math. Soc.**25**(1975), 81-100.**12.**M. E.H. Ismail, On sieved orthogonal polynomials III: Polynomials orthogonal on several intervals, Trans. Amer. Math. Soc.**294**(1986), 89-111.MR**0819937 (87j:33014b)****13.**Mourad E. H. Ismail,*Discriminants and functions of the second kind of orthogonal polynomials*, Results Math.**34**(1998), no. 1-2, 132–149. Dedicated to Paul Leo Butzer. MR**1635590**, https://doi.org/10.1007/BF03322044**14.**Franz Peherstorfer,*On Bernstein-Szegő orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients*, J. Approx. Theory**64**(1991), no. 2, 123–161. MR**1091466**, https://doi.org/10.1016/0021-9045(91)90071-H**15.**F. Peherstorfer and K. Schiefermayr,*Description of extremal polynomials on several intervals and their computation. I, II*, Acta Math. Hungar.**83**(1999), no. 1-2, 27–58, 59–83. MR**1682902**, https://doi.org/10.1023/A:1006607401740**16.**Gábor Szegő,*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517****17.**Ju. Ja. Tomcuk, Orthogonal Polynomials Over a System of Intervals on the Number Line, Zap. Fiz.-Mat. Khar'kov Mat. Oshch.,**29**(1964) 93-128 (in Russian).**18.**V. B. Uvarov,*Relation between polynomials orthogonal with different weights*, Dokl. Akad. Nauk SSSR**126**(1959), 33–36 (Russian). MR**0149187****19.**V. B. Uvarov,*The connection between systems of polynomials that are orthogonal with respect to different distribution functions*, Ž. Vyčisl. Mat. i Mat. Fiz.**9**(1969), 1253–1262 (Russian). MR**0262764**

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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