Elliptic Polynomials: 1st Edition (Hardback) book cover

Elliptic Polynomials

1st Edition

By J.S. Lomont, John Brillhart

Chapman and Hall/CRC

320 pages

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Hardback: 9781584882107
pub: 2000-08-31
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Description

A remarkable interplay exists between the fields of elliptic functions and orthogonal polynomials. In the first monograph to explore their connections, Elliptic Polynomials combines these two areas of study, leading to an interesting development of some basic aspects of each. It presents new material about various classes of polynomials and about the odd Jacobi elliptic functions and their inverses.

The term elliptic polynomials refers to the polynomials generated by odd elliptic integrals and elliptic functions. In studying these, the authors consider such things as orthogonality and the construction of weight functions and measures, finding structure constants and interesting inequalities, and deriving useful formulas and evaluations.

Although some of the material may be familiar, it establishes a new mathematical field that intersects with classical subjects at many points. Its wealth of information on important properties of polynomials and clear, accessible presentation make Elliptic Polynomials valuable to those in real and complex analysis, number theory, and combinatorics, and will undoubtedly generate further research.

Reviews

"…The book under review has several unusual features. Particularly striking is the interplay among mathematical topics for which few connections had been previously noticed. The connection between orthogonal polynomials and their properties with elliptic functions should be valuable to those working in elliptic function. …accessible to a wide variety of mathematicians and students of mathematics. The book serves as an excellent introduction to such topics as orthogonal polynomials, elliptic integrals, and the study of polynomials in general. The authors' exposition is clear with a proper amount of detail. …this monograph is highly recommended.

- Bruce C. Berndt, MathSciNet, American Mathematical Society, Mathematical Reviews on the Web

"…very readable book…material ideal as a spur to undergraduate research, and no less as a gateway to the general study of two venerable subjects not often taught to undergraduates anymore: elliptic functions and orthogonal polynomials."

--D. V. Feldman, University of New Hampshire, CHOICE, October 2001

"The book's strengths lie in its currency, its many worked examples, historical footnotes, and references to the literature."

-D.V. Feldman, University of New Hampshire, in CHOICE, April 1998

Table of Contents

Introduction

Binomial Sequences of Polynomials: The Set of Functions F; The Set of Functions F0

The Sequences {Gm(z)} and {Hm(z)}

The Binomial Sequence Generated from ƒ-1, ƒ F0

The Set of Functions F1 Elliptic Polynomials of the First Kind

The Moment Polynomials Pn(x,y), Qn(x,y), and Rn(x,y)

The Functions F1-1. Elliptic Polynomials of the Second Kind

Inner Products, Integrals, and Moments; Favard's Theorem

The Functions F2. The Orthogonal Sequences {Gm(z)} and {Hm(z)}

The Functions in Classes I and II

The Tangent Numbers

Class III Functions: The n(x) Polynomials.

The Coefficients of the n(x) Polynomials

The n(x) Polynomials

The Orthogonal Sequences {Am(z)} and {Bm(z)}

The Weight Functions for the Sequences {Am(z)} and {Bm(z)}

Miscellaneous Results

Uniqueness and Completion Results

Polynomial Inequalities

Some Concluding Questions

Subject Categories

BISAC Subject Codes/Headings:
MAT022000
MATHEMATICS / Number Theory
MAT036000
MATHEMATICS / Combinatorics
MAT037000
MATHEMATICS / Functional Analysis