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3.2.9 Orthogonal Polynomials

Orthogonal polynomials.

Legendre polynomialsLegendreP[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,,] 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

  • In[1]:= LegendreP[8, x]


  • The integral

    gives zero by virtue of the orthogonality of the Legendre polynomials.
  • In[2]:= Integrate[LegendreP[7,x] LegendreP[8,x], {x, -1, 1}]


  • Integrating the square of a single Legendre polynomial gives a non-zero result.
  • In[3]:= Integrate[LegendreP[8, x]^2, {x, -1, 1}]


  • High-degree Legendre polynomials oscillate rapidly.
  • In[4]:= Plot[LegendreP[10, x], {x, -1, 1}]

  • The associated Legendre "polynomials" contain pieces proportional to

  • In[5]:= LegendreP[8, 3, x]


  • Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
  • In[6]:= LegendreP[8.1, 0]


    Gegenbauer polynomialsGegenbauerC[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.

    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



  • 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.
  • In[7]:= Plot[(HermiteH[6, x] Exp[-x^2/2])^2, {x, -6, 6}]

    Generalized Laguerre polynomialsLaguerreL[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 .
    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 and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials. The Jacobi polynomials are sometimes given in the alternative form


  • You can get formulas for generalized Laguerre polynomials with arbitrary values of a.
  • In[8]:= LaguerreL[2, a, x]