Chebyshev-like polynomials, conic distribution of roots, and continued fractions

Oyengo, Michael Obiero

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https://hdl.handle.net/2142/101135 Copy

Description

Title

Chebyshev-like polynomials, conic distribution of roots, and continued fractions

Author(s)

Oyengo, Michael Obiero

Issue Date

2018-04-02

Director of Research (if dissertation) or Advisor (if thesis)

Stolarsky, Kenneth B.

Doctoral Committee Chair(s)

Berndt, Bruce C.

Committee Member(s)

• Hildebrand, A.J. J
• Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Chebyshev polynomials
• Conics
• Chebyshev points
• Chebyshev-like polynomials
• root distribution
• rationally generated polynomials
• polynomials with bi-concyclic roots
• periodic continued fractions
• quadratic irrationals
• non-periodic continued fractions
• large partial quotients
• integer multiples of periodic continued fractions

Abstract

In the first part of this thesis, we show that a wide range of the properties of the roots of translated Chebyshev polynomials of the first kind (call these complex numbers Chebyshev points), are illuminated by the study of geometric properties of the ellipse and conversely. This includes the following; Chebyshev points lie on certain ellipses centered at the origin, and conversely, every such ellipse contains infinitely many sets of Chebyshev points. In special cases, the products of all curvatures at Chebyshev points can be expressed in terms of Fibonacci and Lucas numbers. Sets of certain Chebyshev points interlace on an ellipse. We also show that Chebyshev points generated by Chebyshev polynomials of the second kind lie very close to certain high order algebraic curves that are ``nearly'' ellipses. Next, we examine roots of linear combinations of Chebyshev polynomials. Here we use continued fractions to give estimates for the roots that do not lie in the interval $(-1,1)$. We then show the connection between polynomials with roots on concentric circles to polynomials with roots on ellipses. In particular, we construct a sequence of polynomials satisfying a fourth-order recurrence relation with a parameter $c$, by replacing coefficients of finite geometric series by Chebyshev polynomials in a simple way. We show that for $c$ real, the roots of these polynomials lie on two concentric but inversely related circles. The associated $n\times n$ Hankel determinants are also determined. We conclude the first part of the thesis by studying some polynomials that are related to Chebyshev polynomials, and having their roots on hyperbolas. In the second part of the thesis, we study continued fractions of quadratic irrationals. We construct a class of rationals and quadratic irrationals having continued fractions whose period has length $n\geq2$, and with ``small'' partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with ``large'' partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of these continued fractions and show that polynomials arising from these generalizations are related to Chebyshev and Fibonacci polynomials. Finally, we examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases, they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.

Graduation Semester

2018-05

Type of Resource

text

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http://hdl.handle.net/2142/101135

Copyright and License Information

Copyright 2018 Michael Obiero Oyengo

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