# Orthogonal Polynomials

LegendreP[n,x] | Legendre polynomials |

LegendreP[n,m,x] | associated Legendre polynomials |

SphericalHarmonicY[l,m,θ,ϕ] | 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 |

*Legendre polynomials* LegendreP[n,x] 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[n,m,x] 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[l,m,θ,ϕ] are related to associated Legendre polynomials. They satisfy the orthogonality relation for or , where represents integration over the surface of the unit sphere.

*Gegenbauer polynomials* GegenbauerC[n,m,x] can be viewed as generalizations of the Legendre polynomials to systems with ‐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 . 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[n,z] 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[n,x] 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 .

*Generalized Laguerre polynomials* LaguerreL[n,a,x] are related to hydrogen atom wave functions in quantum mechanics. They satisfy the differential equation , and the orthogonality relation for . The *Laguerre polynomials* LaguerreL[n,x] correspond to the special case .

*Zernike radial polynomials* ZernikeR[n,m,x] are used in studies of aberrations in optics. They satisfy the orthogonality relation for .

*Jacobi polynomials* JacobiP[n,a,b,x] 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 .