Elliptic Polynomials / Edition 1

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Overview

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.

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Editorial Reviews

Booknews
This book explores the interplay between the fields of elliptic functions and orthogonal polynomials, considering such topics as orthogonality and the construction of weight functions and measures, finding structure constants, and deriving useful formulas and evaluations. It also provides new material about Jacobi elliptic functions through the use of Maclaurin series divisibility, and introduces a new class of polynomials, "moment polynomials", which give the moments of orthogonal sequences. Lomont and Brillhart (both U. of Arizona, Tucson) have written this text for people involved in real and complex analysis, number theory and combinatorics. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9781584882107
  • Publisher: Taylor & Francis
  • Publication date: 8/31/2000
  • Edition number: 1
  • Pages: 320
  • Product dimensions: 6.40 (w) x 9.50 (h) x 0.91 (d)

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

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