Orthogonal Polynomial

orthogonal polynomial

[ȯr′thäg·ən·əl päl·ə′nō·mē·əl]
Orthogonal polynomials are various families of polynomials, which arise as solutions to differential equations related to the hypergeometric equation, and which are mutually orthogonal as functions.

Orthogonal Polynomial


A set of orthogonal polynomials is a special system of polynomials {pn(x)}, n = 0, 1, 2, …, that are orthogonal with respect to some weight function ρ(x) on an interval [a, b]. A normalized system of orthogonal polynomials is denoted by n(x), and a system of orthogonal polynomials whose leading coefficients are all equal to 1 is denoted by n (x). In boundary value problems of mathematical physics, systems of orthogonal polynomials are often encountered where the weight function ρ(x) satisfies the following differential equation (Pearson’s equation):

A polynomial pn (x) in such a system satisfies the differential equation

where λn = n[α1 + (n + 1) β2].

The most important classical systems of orthogonal polynomials are of this type. Aside from constant multipliers, the polynomials are determined by the choices of a, b, and ρ(x) indicated below.

(1) Jacobi polynomials, {pn(λ,μ)(x)}, correspond to a = –1, b = 1, ρ(x) = (1 – x)λ (1 + x)μ, λ > –1, and μ > –1. Certain values of λ and μ are associated with special cases of Jacobi polynomials. Thus, for λ = μ we have the ultraspherical polynomials, Pn(λ)(x), which are sometimes called Gegen-bauer polynomials. When λ = μ = –½, that is, ρ(x) = Orthogonal Polynomial we have the Chebyshev polynomials of the first kind, Tn(x); and for λ = μ = ½, that is Orthogonal Polynomial, we have the Chebyshev polynomials of the second kind, Un(x). For λ = μ = 0, that is, ρ(x) ≡ 1, we have the Legendre polynomials, Pn(x).

(2) Laguerre polynomials, Ln (x), correspond to a = 0, b = + ∞, and ρ(x) = e–x. For ρ(x) = xae–x >–1), we have the generalized Laguerre polynomials, Orthogonal Polynomial.

(3) Hermite polynomials, Hn(x), for a = –, ∞, b = + ∞, and

ρ(x) = ex2

Orthogonal polynomials have many properties in common. The zeroes of the polynomials pn(x) are real and simple and are located within the interval [a, b]. Between any two successive zeroes of pn (x) there lies a zero of pn+1 (x). The polynomial pn (x) is given by Rodrigues’ formula

where An is a constant and β(x) is of the form given in (*) above. Our systems of orthogonal polynomials are complete. Three successive orthogonal polynomials n(x), n + 1 (x), and n + 2 (x) are related by the recursion formula

n + 2 (x) = (xαn + 2) n + 1)(xλn + 1(x)

where αn + 2 and λn + 1 are given by the coefficients of the polynomials as follows: if


αn + 2 = αn + 1,nαn + 2, n + 1

λn + 1 = αn + 1, n – 1αn + 2αn + 1, nαn + 2, n

P. L. Chebyshev developed a general theory of orthogonal polynomials. His basic method of studying orthogonal polynomials was to expand the integral

in a continued fraction with partial denominators xαn and partial numerators λn – 1. The denominators of the convergents ϕn(x)/pn (x) of this continued fraction form a system of orthogonal polynomials on the interval [a, b] with respect to the weight function ρ(x).

The classical systems of orthogonal polynomials given above can be expressed in terms of the hypergeometric function.


Szegö, G. Ortogonal’nye mnogochleny. Moscow, 1962. (Translated from English.)


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