Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

Details Nonlinear Integral-Equation Formulation of Orthogonal Polynomials Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

2006-10-26

arxiv.org/abs/math-ph/0610071v2

J. Phys. A 40, F9 (2007)

Physics

Mathematical Physics

7 pages, result generalized to include integration in the complex domain

Scientific paper

10.1088/1751-8113/40/1/F02

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.

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