Elliptic Polynomials

Lomont, J.S.; Brillhart, John

The interplay between the fields of elliptic functions and orthogonal polynomials produces interesting developments in the basic aspects of each. Elliptic Polynomials is the first monograph to explore the connections between the two. The authors furnish many new formulas on special functions and polynomials and offer a wealth of information regarding the important properties of polynomials.

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 ¿-1, where ¿ 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