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Table of Integrals, Series, and Products, 8th Edition by Daniel Zwillinger

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8.9 Orthogonal Polynomials

8.90 Introduction

8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, …, the integral

abxnw(x)dx

si1530_e

exists and that the integral

abw(x)dx

si1531_e

is positive. In this case, there exists a sequence of polynomials p0(x), p1(x), …, pn(x), …, that is uniquely determined by the following conditions:

1. pn(x) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.

2. The polynomials ...

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