Elliptic Polynomials by J.S. Lomont & John Brillhart - Scholarly

In this review of Elliptic Polynomials, the authors J.S. Lomont and John Brillhart present a focused monograph aimed at researchers and advanced students who want a bridge between elliptic functions and orthogonal polynomials. The bottom line: this is a specialized, rigorous treatment best suited for mathematicians interested in the structural links and explicit constructions that arise when odd Jacobi elliptic functions generate polynomial families. The book's chief draw is the novel development of weight functions, measures and identities that arise from elliptic integrals.

Key Features

• Interdisciplinary focus: Explores the interplay between elliptic functions and orthogonal polynomials, giving readers a coherent narrative linking the two areas.
• New material: Presents original results about classes of polynomials generated by odd elliptic integrals, useful for further theoretical work.
• Orthogonality construction: Develops weight functions and measures explicitly, helping readers apply these polynomials in analytic contexts.
• Technical evaluations: Provides structure constants and inequalities that support rigorous estimates and further derivations.
• Function inverses: Covers odd Jacobi elliptic functions and their inverses, clarifying inversion relations used in constructing polynomial families.

Who It's For

Elliptic Polynomials is primarily for professional mathematicians, graduate students in pure mathematics, and researchers in special functions who already have familiarity with elliptic functions or orthogonal polynomial theory. The treatment assumes comfort with advanced analysis and classical function theory rather than introductory exposition.

Readers seeking a textbook-style introduction or applied numerical algorithms should look elsewhere, since this monograph emphasizes structural results, explicit formulas, and theoretical connections over pedagogical progression or computational recipes.

Pros & Cons

Pros

• Clear presentation of the link between elliptic integrals and polynomial generation, valuable for theoretical development.
• Concrete construction of weight functions and measures that illuminate orthogonality properties.
• New inequalities and structure constants that make subsequent work and comparisons easier.

Cons

• The material is specialized and assumes significant background, so it is not suited to beginners seeking an introduction.

Specifications

Title	Elliptic Polynomials
Authors	J.S. Lomont, John Brillhart
Subject	Elliptic functions and orthogonal polynomials
Coverage	Odd Jacobi elliptic functions, inverses, weight functions
Includes	Structure constants, inequalities, explicit formulas
Audience	Researchers and advanced graduate students in mathematics

Our Verdict

Elliptic Polynomials is a valuable, tightly focused monograph for those working at the intersection of special functions and orthogonal polynomials; its explicit constructions and inequalities make it good value for researchers who will use the formulas and measures in further theoretical work.

Frequently Asked Questions

Does this book introduce elliptic functions from scratch?
Answer. No, the text assumes prior familiarity with elliptic functions and moves quickly into connections with polynomial constructions.

Are computational algorithms provided?
Answer. The emphasis is theoretical: explicit formulas and evaluations appear, but practical numerical algorithms are not the focus.

Who benefits most from reading this book?
Answer. Researchers in pure mathematics and advanced graduate students studying special functions or orthogonal polynomials will benefit most.