# Orthogonal Polynomial

## orthogonal polynomial

[ȯr′thäg·ən·əl päl·ə′nō·mē·əl]## Orthogonal Polynomial

A set of orthogonal polynomials is a special system of polynomials {*p _{n}(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

*p̂*), and a system of orthogonal polynomials whose leading coefficients are all equal to 1 is denoted by

_{n}(x*p̂*(

_{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 *p _{n}* (

*x*) in such a system satisfies the differential equation

where *λ _{n}* =

*n[α*

_{1}+ (

*n*+ 1)

*β*

_{2}].

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, {*p _{n}*

^{(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,

*P*

_{n}^{(λ)}(

*x*), which are sometimes called Gegen-bauer polynomials. When

*λ*=

*μ*= –½, that is,

*ρ(x*) = we have the Chebyshev polynomials of the first kind,

*T*); and for

_{n}(x*λ*=

*μ*= ½, that is , we have the Chebyshev polynomials of the second kind,

*U*). For

_{n}(x*λ*=

*μ*= 0, that is,

*ρ(x*) ≡ 1, we have the Legendre polynomials,

*P*).

_{n}(x(2) Laguerre polynomials, *L _{n}* (

*x*), correspond to

*a*= 0,

*b*= + ∞, and

*ρ(x*) =

*e*. For

^{–x}*ρ(x*) =

*x*>

^{a}e^{–x}(α^{–1}), we have the generalized Laguerre polynomials, .

(3) Hermite polynomials, *H _{n}(x*), for

*a*= –, ∞,

*b = + ∞*, and

*ρ(x*) = *e*^{–x2}

Orthogonal polynomials have many properties in common. The zeroes of the polynomials *p _{n}(x*) are real and simple and are located within the interval [

*a, b*]. Between any two successive zeroes of

*p*(

_{n}*x*) there lies a zero of

*p*

_{n+1}(

*x*). The polynomial

*p*(

_{n}*x*) is given by Rodrigues’ formula

where *A _{n}* is a constant and

*β(x*) is of the form given in (*) above. Our systems of orthogonal polynomials are complete. Three successive orthogonal polynomials

*p̃*),

_{n}(x*p̃*

_{n + 1}(

*x*), and

*p̃*

_{n + 2}(

*x*) are related by the recursion formula

*p̃*_{n + 2} (*x*) = (*x* – *α*_{n + 2}) *p̃*_{n + 1})(*x* – *λ*_{n + 1}(*x*)

where *α*_{n + 2} and λ_{n + 1} are given by the coefficients of the polynomials as follows: if

then

*α*_{n + 2} = *α*_{n + 1,n} – *α*_{n + 2, n + 1}

*λ*_{n + 1} = *α*_{n + 1, n – 1} – *α*_{n + 2}*α*_{n + 1, n} – *α*_{n + 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 *x*—*α _{n}* and partial numerators

*λ*

_{n – 1}. The denominators of the convergents ϕ

_{n}(

*x*)/

*p*(

_{n}*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.

### REFERENCE

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

V. I. BITIUTSKOV