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Orthogonal Polynomials

LegendreP[n,x]Legendre polynomials Pn (x)
LegendreP[n,m,x]associated Legendre polynomials
SphericalHarmonicY[l,m,,]spherical harmonics
GegenbauerC[n,m,x]Gegenbauer polynomials
ChebyshevT[n,x], ChebyshevU[n,x]Chebyshev polynomials Tn (x) and Un (x) of the first and second kinds
HermiteH[n,x]Hermite polynomials Hn (x)
LaguerreL[n,x]Laguerre polynomials Ln (x)
LaguerreL[n,a,x]generalized Laguerre polynomials
ZernikeR[n,m,x] Zernike radial polynomials
JacobiP[n,a,b,x]Jacobi polynomials

Orthogonal polynomials.

Legendre polynomials LegendreP[n, x] arise in studies of systems with three-dimensional spherical symmetry. They satisfy the differential equation (1-x2)y-2xy+n (n+1)y=0, and the orthogonality relation for mn.
The associated Legendre polynomials LegendreP[n, m, x] are obtained from derivatives of the Legendre polynomials according to . Notice that for odd integers mn, the contain powers of , and are therefore not strictly polynomials. The reduce to Pn (x) when m=0.
The spherical harmonics SphericalHarmonicY[l, m, , ] are related to associated Legendre polynomials. They satisfy the orthogonality relation for ll or mm, where d represents integration over the surface of the unit sphere.
This gives the algebraic form of the Legendre polynomial P8 (x).
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The integral gives zero by virtue of the orthogonality of the Legendre polynomials.
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Integrating the square of a single Legendre polynomial gives a non-zero result.
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High-degree Legendre polynomials oscillate rapidly.
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The associated Legendre "polynomials" involve fractional powers.
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"Special Functions" discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
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Gegenbauer polynomials 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.
GegenbauerC[n, 0, x] is always equal to zero. GegenbauerC[n, x] is however given by the limit . This form is sometimes denoted .
Series of Chebyshev polynomials are often used in making numerical approximations to functions. The Chebyshev polynomials of the first kind ChebyshevT[n, x] are defined by Tn (cos)=cos (n). They are normalized so that Tn (1)=1. They satisfy the orthogonality relation for mn. The Tn (x) also satisfy an orthogonality relation under summation at discrete points in x corresponding to the roots of Tn (x).
The Chebyshev polynomials of the second kind ChebyshevU[n, z] are defined by Un (cos)=sin[ (n+1)]/sin. With this definition, Un (1)=n+1. The Un satisfy the orthogonality relation for mn.
The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.
Hermite polynomials 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 for mn. An alternative form of Hermite polynomials sometimes used is (a different overall normalization of the Hen (x) is also sometimes used).
The Hermite polynomials are related to the parabolic cylinder functions or Weber functions Dn (x) by .
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.
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Generalized Laguerre polynomials 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 for mn. The Laguerre polynomials LaguerreL[n, x] correspond to the special case a=0.
You can get formulas for generalized Laguerre polynomials with arbitrary values of a.
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Zernike radial polynomials ZernikeR[n, m, x] are used in studies of aberrations in optics. They satisfy the orthogonality relation for nk.
Jacobi polynomials JacobiP[n, a, b, x] occur in studies of the rotation group, particularly in quantum mechanics. They satisfy the orthogonality relation for mn. 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 .