Orthogonal Polynomials

Orthogonal polynomials.

Legendre polynomials (RefLink[LegendreP, paclet:ref/LegendreP])[n,x] arise in studies of systems with three-dimensional spherical symmetry. They satisfy the differential equation (1-(x^2))y''-2xy'+n(n+1)y==0, and the orthogonality relation (∫_(-1)^1)(Subscript[P, m])(x)(Subscript[P, n])(x)dx==0 for m≠n.

The associated Legendre polynomials (RefLink[LegendreP, paclet:ref/LegendreP])[n,m,x] are obtained from derivatives of the Legendre polynomials according to (Subscript[P, n]^m)(x)==(((-1))^m)(((1-(x^2)))^(m/2)) (d^m)[(Subscript[P, n])(x)]/d(x^m). Notice that for odd integers m≤n, the (Subscript[P, n]^m)(x) contain powers of (Sqrt[1-(x^2)]), and are therefore not strictly polynomials. The (Subscript[P, n]^m)(x) reduce to (Subscript[P, n])(x) when m==0.

The spherical harmonics (RefLink[SphericalHarmonicY, paclet:ref/SphericalHarmonicY])[l,m,θ,ϕ] are related to associated Legendre polynomials. They satisfy the orthogonality relation ∫(Subscript[Y, l]^( m))(θ,ϕ)(Subscript[(OverBar[Y]), l']^( m'))(θ,ϕ)dω==0 for l≠l' or m≠m', where dω represents integration over the surface of the unit sphere.

This gives the algebraic form of the Legendre polynomial (Subscript[P, 8])(x):

LegendreP[8,x]

The integral (∫_(-1)^1)(Subscript[P, 7])(x) (Subscript[P, 8])(x) dx gives zero by virtue of the orthogonality of the Legendre polynomials:

Integrate[LegendreP[7,x] LegendreP[8,x],{x,-1,1}]

Integrating the square of a single Legendre polynomial gives a nonzero result:

Integrate[LegendreP[8,x]^2,{x,-1,1}]

High-degree Legendre polynomials oscillate rapidly:

Plot[LegendreP[10,x],{x,-1,1}]

The associated Legendre "polynomials" involve fractional powers:

LegendreP[8,3,x]

"Special Functions" discusses the generalization of Legendre polynomials to Legendre functions, which can have noninteger degrees:

LegendreP[8.1,0]

Gegenbauer polynomials (RefLink[GegenbauerC, paclet:ref/GegenbauerC])[n,m,x] can be viewed as generalizations of the Legendre polynomials to systems with (m+2)-dimensional spherical symmetry. They are sometimes known as ultraspherical polynomials.

(RefLink[GegenbauerC, paclet:ref/GegenbauerC])[n,0,x] is always equal to zero. (RefLink[GegenbauerC, paclet:ref/GegenbauerC])[n,x] is however given by the limit (lim_(m->0))(Subscript[C, n]^((m)))(x)/m. This form is sometimes denoted (Subscript[C, n]^((0)))(x).

Series of Chebyshev polynomials are often used in making numerical approximations to functions. The Chebyshev polynomials of the first kind (RefLink[ChebyshevT, paclet:ref/ChebyshevT])[n,x] are defined by (Subscript[T, n])(cosθ)==cos(nθ). They are normalized so that (Subscript[T, n])(1)==1. They satisfy the orthogonality relation (∫_(-1)^1)(Subscript[T, m])(x)(Subscript[T, n])(x)(((1-(x^2)))^(-1/2))dx==0 for m≠n. The (Subscript[T, n])(x) also satisfy an orthogonality relation under summation at discrete points in x corresponding to the roots of (Subscript[T, n])(x).

The Chebyshev polynomials of the second kind (RefLink[ChebyshevU, paclet:ref/ChebyshevU])[n,z] are defined by (Subscript[U, n])(cosθ)==sin[(n+1)θ]/sinθ. With this definition, (Subscript[U, n])(1)==n+1. The (Subscript[U, n]) satisfy the orthogonality relation (∫_(-1)^1)(Subscript[U, m])(x)(Subscript[U, n])(x)(((1-(x^2)))^(1/2))dx==0 for m≠n.

The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.

Hermite polynomials (RefLink[HermiteH, paclet:ref/HermiteH])[n,x] arise as the quantum-mechanical wave functions for a harmonic oscillator. They satisfy the differential equation y''-2xy'+2ny==0, and the orthogonality relation (∫_(-∞)^∞)(Subscript[H, m])(x)(Subscript[H, n])(x)(e^(-(x^2)))dx==0 for m≠n. An alternative form of Hermite polynomials sometimes used is (Subscript[He, n])(x)==(2^(-n/2))(Subscript[H, n])(x/(Sqrt[2])) (a different overall normalization of the (Subscript[He, n])(x) is also sometimes used).

The Hermite polynomials are related to the parabolic cylinder functions or Weber functions (Subscript[D, n])(x) by (Subscript[D, n])(x)==(2^(-n/2))(e^(-(x^2)/4))(Subscript[H, n])(x/(Sqrt[2])).

This gives the density for an excited state of a quantum-mechanical harmonic oscillator. The average of the wiggles is roughly the classical physics result:

Plot[(HermiteH[6,x] Exp[-x^2/2])^2,{x,-6,6}]

Generalized Laguerre polynomials (RefLink[LaguerreL, paclet:ref/LaguerreL])[n,a,x] are related to hydrogen atom wave functions in quantum mechanics. They satisfy the differential equation xy''+(a+1-x)y'+ny==0, and the orthogonality relation (∫_0^∞)(Subscript[L, m]^a)(x)(Subscript[L, n]^a)(x)(x^a)(e^(-x))dx==0 for m≠n. The Laguerre polynomials (RefLink[LaguerreL, paclet:ref/LaguerreL])[n,x] correspond to the special case a==0.

You can get formulas for generalized Laguerre polynomials with arbitrary values of a:

LaguerreL[2,a,x]

Zernike radial polynomials (RefLink[ZernikeR, paclet:ref/ZernikeR])[n,m,x] are used in studies of aberrations in optics. They satisfy the orthogonality relation (∫_0^1)(Subscript[R, n]^((m)))(x)(Subscript[R, k]^((m)))(x)xdx==0 for n≠k.

Jacobi polynomials (RefLink[JacobiP, paclet:ref/JacobiP])[n,a,b,x] occur in studies of the rotation group, particularly in quantum mechanics. They satisfy the orthogonality relation (∫_(-1)^1)(Subscript[P, m]^((a,b)))(x)(Subscript[P, n]^((a,b)))(x)(((1-x))^a)(((1+x))^b)dx==0 for m≠n. Legendre, Gegenbauer, Chebyshev and Zernike polynomials can all be viewed as special cases of Jacobi polynomials. The Jacobi polynomials are sometimes given in the alternative form (Subscript[G, n])(p,q,x)==n! Γ(n+p)/Γ(2n+p) (Subscript[P, n]^((p-q,q-1)))(2x-1).