Table of Integrals, Series, and Products, 8th Edition

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

${\displaystyle {\int }_a^b{x}^{n}w(x)\text{d}x}$

exists and that the integral

${\displaystyle {\int }_a^{b}w(x)\text{d}x}$

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

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

2. The polynomials ...