Elliptic Polynomials / Edition 1

by J.S. Lomont, John Brillhart

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.

Product Details

ISBN-13:	9781584882107
Publisher:	Taylor & Francis
Publication date:	08/31/2000
Pages:	320
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Lomont, J.S.; Brillhart, John

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 Derived from ƒ-1, where ƒ F0;The Set of Functions F1 Elliptic Integrals and Polynomials of the First Kind; The Moment Polynomials Pn(x,y), Qn(x,y), and Rn(x,y); The Set of Functions F1-1; Elliptic Functions and Polynomials of the Second Kind; Inner Products, Integrals, and Moments. Favard's Theorem; The Set of Functions F2. The Orthogonal Sequences {Gm(z)} and {Hm(z)}; Class I Functions. Class II Functions, Tangent Numbers, Class III Functions: The Modified Mittag-Leffler Polynomial n(x). Structure Constants; The Coefficients of the n(x) Polynomials, The n(x) Polynomials, The Orthogonal Sequences {Am(z)} and {Bm(z)} . Structure Constants; Weight Functions for the Sequences {Am(z)} and {Bm(z)}; Miscellaneous Results, Uniqueness and Completion Results, Polynomial Inequalities, Some Concluding Questions

Editorial Reviews

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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