Orthogonal Polynomials
| LegendreP[n,x] | Legendre polynomials  |
| 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  and  of the first and second kinds |
| HermiteH[n,x] | Hermite polynomials  |
| LaguerreL[n,x] | Laguerre polynomials  |
| 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
arise in studies of systems with three-dimensional spherical symmetry. They satisfy the differential equation
, and the orthogonality relation
for
.
The associated Legendre polynomials
LegendreP
are obtained from derivatives of the Legendre polynomials according to
. Notice that for odd integers
, the
contain powers of
, and are therefore not strictly polynomials. The
reduce to
when
.
The spherical harmonics
SphericalHarmonicY
are related to associated Legendre polynomials. They satisfy the orthogonality relation
for
or
, where
represents integration over the surface of the unit sphere.
This gives the algebraic form of the Legendre polynomial
.
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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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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 noninteger degrees.
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Gegenbauer polynomials
GegenbauerC
can be viewed as generalizations of the Legendre polynomials to systems with
-dimensional spherical symmetry. They are sometimes known as ultraspherical polynomials.
GegenbauerC
is always equal to zero.
GegenbauerC
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
are defined by
. They are normalized so that
. They satisfy the orthogonality relation
for
. The
also satisfy an orthogonality relation under summation at discrete points in
corresponding to the roots of
.
The Chebyshev polynomials of the second kind
ChebyshevU
are defined by
. With this definition,
. The
satisfy the orthogonality relation
for
.
The name "Chebyshev" is a transliteration from the Cyrillic alphabet; several other spellings, such as "Tschebyscheff", are sometimes used.
Hermite polynomials
HermiteH
arise as the quantum-mechanical wave functions for a harmonic oscillator. They satisfy the differential equation
, and the orthogonality relation
for
. An alternative form of Hermite polynomials sometimes used is
(a different overall normalization of the
is also sometimes used).
The Hermite polynomials are related to the parabolic cylinder functions or Weber functions
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
are related to hydrogen atom wave functions in quantum mechanics. They satisfy the differential equation
, and the orthogonality relation
for
. The Laguerre polynomials
LaguerreL
correspond to the special case
.
You can get formulas for generalized Laguerre polynomials with arbitrary values of
.
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Zernike radial polynomials
ZernikeR
are used in studies of aberrations in optics. They satisfy the orthogonality relation
for
.
Jacobi polynomials
JacobiP
occur in studies of the rotation group, particularly in quantum mechanics. They satisfy the orthogonality relation
for
. 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
.
