Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
摘要
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.
引用
@article{arxiv.math-ph/0610071,
title = {Nonlinear Integral-Equation Formulation of Orthogonal Polynomials},
author = {Carl M. Bender and E. Ben-Naim},
journal= {arXiv preprint arXiv:math-ph/0610071},
year = {2007}
}
备注
7 pages, result generalized to include integration in the complex domain