Orthogonal Polynomials

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August 2007

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371361D-01

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A set of polynomials P_i(x) are orthogonal polynomials over the interval a < x < b if each polynomial in the set satisfies the following equations.

(A)

The interval (a, b) and the weighting function w(x) vary depending on the set of orthogonal polynomials. One of the most important applications of orthogonal polynomials is to solve differential equations.

Chebyshev Orthogonal Polynomials of the First Kind

The recurrence relationship defines Chebyshev orthogonal polynomials of the first kind T_n(x), as shown by the following equations.

T₀(x) = 1

(B)

T₁(x) = x

(C)

T_n(x) = 2xT_n_{– 1}(x) – T_n_{– 2}(x)
for n = 2, 3, …

(D)

Chebyshev orthogonal polynomials of the first kind satisfy the following equations.

(E)

Chebyshev Orthogonal Polynomials of the Second Kind

The recurrence relationship defines Chebyshev orthogonal polynomials of the second kind U_n(x), as shown by the following equations.

U₀(x) = 1

(F)

U₁(x) = 2x

(G)

U_n(x) = 2xU_n_{– 1}(x) – U_n_{– 2}(x)
for n = 2, 3, …

(H)

Chebyshev orthogonal polynomials of the second kind satisfy the following equations.

(I)

Gegenbauer Orthogonal Polynomials

The recurrence relationship defines Gegenbauer orthogonal polynomials C_n^a(x) as shown by the following equations.

C₀^a(x) = 1

(J)

C₁^a(x) = 2ax

(K)

(L)

Gegenbauer orthogonal polynomials satisfy the following equations.

(M)

where Γ(z) is a gamma function defined by the following equation.

(N)

Hermite Orthogonal Polynomials

The recurrence relationship defines Hermite orthogonal polynomials H_n(x), as shown by the following equations.

H₀(x) = 1

(O)

H₁(x) = 2x

(P)

H_n(x) = 2xH_n_{– 1}(x) – 2(n – 1)H_n_{– 2}(x)
for n = 2, 3, …

(Q)

Hermite orthogonal polynomials satisfy the following equations.

(R)

Laguerre Orthogonal Polynomials

The recurrence relationship defines Laguerre orthogonal polynomials L_n(x), as shown by the following equations.

L₀(x) = 1

(S)

L₁(x) = –x + 1

(T)

(U)

Laguerre orthogonal polynomials satisfy the following equations.

(V)

Associated Laguerre Orthogonal Polynomials

The recurrence relationship defines associated Laguerre orthogonal polynomials L_n^a(x) as shown by the following equations.

L₀^a(x) = 1

(W)

L₁^a(x) = –x + a + 1

(X)

(Y)

Associated Laguerre orthogonal polynomials satisfy the following equation.

(Z)

Legendre Orthogonal Polynomials

The recurrence relationship defines Legendre orthogonal polynomials P_n(x), as shown by the following equations.

P₀(x) = 1

(AA)

P₁(x) = x

(AB)

(AC)

Legendre orthogonal polynomials satisfy the following equations.

(AD)