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_n(x), and a system of orthogonal polynomials whose leading coefficients are all equal to 1 is denoted by 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[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, {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_n(x); and for λ = μ = ½, that is , we have the Chebyshev polynomials of the second kind, U_n(x). For 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^–x. For ρ(x) = x^ae^–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.