Nonlinear-integral-equation construction of orthogonal polynomials

Abstract

The nonlinear integral equation P(x) = ∫αβ dyw(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 Pn(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 xw(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. Finally, it is observed that since the integral equation is independent of the degree of the polynomials it may possibly be a useful tool in determining and studying the asymptotic behaviors of polynomials. Copyright © 2008 by Carl M Bender and E Ben-Naim.