Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

TL;DR

This paper introduces a nonlinear integral equation that characterizes all orthogonal polynomials for a given weight function, offering a new approach to extend orthogonal polynomial theory into the complex domain.

Contribution

It presents a novel nonlinear integral equation formulation that generates orthogonal polynomials and simplifies their analysis and extension into complex domains.

Findings

Polynomial solutions form an orthogonal set with respect to x w(x)

The integral equation reduces to linear algebraic equations for polynomial coefficients

The approach simplifies the specification and extension of orthogonal polynomials

Abstract

The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure x w(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.