# 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

.

Out[1]= | |

The integral

gives zero by virtue of the orthogonality of the Legendre polynomials.

Out[2]= | |

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

Out[3]= | |

High-degree Legendre polynomials oscillate rapidly.

Out[4]= | |

The associated Legendre "polynomials" involve fractional powers.

Out[5]= | |

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

Out[6]= | |

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.

Out[7]= | |

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

.

Out[8]= | |

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

.