Orthogonal Polynomials
LegendreP[n,x]  Legendre polynomials P_{n} (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 T_{n} (x) and U_{n} (x) of the first and second kinds 
HermiteH[n,x]  Hermite polynomials H_{n} (x) 
LaguerreL[n,x]  Laguerre polynomials L_{n} (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 threedimensional spherical symmetry. They satisfy the differential equation
(1x^{2})y^{}2xy^{}+n (n+1)y=0, and the orthogonality relation
for
m≠n.
The associated Legendre polynomials
LegendreP[n, m, x] are obtained from derivatives of the Legendre polynomials according to
. Notice that for odd integers
m≤n, the
contain powers of
, and are therefore not strictly polynomials. The
reduce to
P_{n} (x) when
m=0.
The spherical harmonics
SphericalHarmonicY[l, m, , ] are related to associated Legendre polynomials. They satisfy the orthogonality relation
for
l≠l^{} or
m≠m^{}, where
d represents integration over the surface of the unit sphere.
This gives the algebraic form of the Legendre polynomial P_{8} (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 nonzero result.
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Highdegree 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 noninteger 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
T_{n} (cos)=cos (n). They are normalized so that
T_{n} (1)=1. They satisfy the orthogonality relation
for
m≠n. The
T_{n} (x) also satisfy an orthogonality relation under summation at discrete points in
x corresponding to the roots of
T_{n} (x).
The Chebyshev polynomials of the second kind
ChebyshevU[n, z] are defined by
U_{n} (cos)=sin[ (n+1)]/sin. With this definition,
U_{n} (1)=n+1. The
U_{n} satisfy the orthogonality relation
for
m≠n.
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 quantummechanical wave functions for a harmonic oscillator. They satisfy the differential equation
y^{}2xy^{}+2ny=0, and the orthogonality relation
for
m≠n. An alternative form of Hermite polynomials sometimes used is
(a different overall normalization of the
He_{n} (x) is also sometimes used).
The Hermite polynomials are related to the parabolic cylinder functions or Weber functions
D_{n} (x) by
.
This gives the density for an excited state of a quantummechanical 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+1x)y^{}+ny=0, and the orthogonality relation
for
m≠n. 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
n≠k.
Jacobi polynomials
JacobiP[n, a, b, x] occur in studies of the rotation group, particularly in quantum mechanics. They satisfy the orthogonality relation
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
.