Other Orthogonal Polynomials

Abstract

5.1 Introduction A set of functions {ϕ n (x)},n=0,1,2,…, is said to be orthogonal on the interval a<x<b , with respect to a weight function r(x)<0 , if ∫ b a r(x)ϕ n (x)ϕ k (x)dx=0k≠n Sets of orthogonal functions play an extremely important role in analysis, primarily because functions belonging to a very general class can be represented by series of orthogonal functions, called generalized Fourier series. A special class of orthogonal functions consists of the sets of orthogonal polynomials{p n (x)} , where n denotes the degree of the polynomial p n (x) . The Legendre polynomials discussed in Chap. 4 are probably the simplest set of polynomials belonging to this class. Other polynomial sets which commonly occur in applications are the Hermite, Laguerre, and Chebyshev polynomials. More general polynomial sets are defined by the Gegenbauer and Jacobi polynomials, which include the others as special cases. The study of general polynomial sets like the Jacobi polynomials facilitates the study of each polynomial set by focusing on those properties that are characteristic of all the individual sets.