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Maxima Manual

16.2 Definitions for Orthogonal Polynomials

Function: ASSOC_LEGENDRE_P (n, m, x)

[specfun package] return the associated Legendre function of the first kind for integers n > -1 and m > -1. When | m | > n and n >= 0, we have assoc_legendre_p (n, m, x) = 0. Reference: A&S 22.5.37 page 779, A&S 8.6.6 (second equation) page 334, and A&S 8.2.5 page 333. To access this function, load("specfun"). See ASSOC_LEGENDRE_Q, LEGENDRE_P, and LEGENDRE_Q.

Function: ASSOC_LEGENDRE_Q (n, m, x)

[specfun package] return the associated Legendre function of the second kind for integers n > -1 and m > -1.

Reference: Gradshteyn and Ryzhik 8.706 page 1000.

To access this function, load("specfun").

See also ASSOC_LEGENDRE_P, LEGENDRE_P, and LEGENDRE_Q.

Function: CHEBYSHEV_T (n, x)

[specfun package] return the Chebyshev function of the first kind for integers n > -1.

Reference: A&S 22.5.31 page 778 and A&S 6.1.22 page 256.

To access this function, load("specfun").

See also CHEBYSHEV_U.

Function: CHEBYSHEV_U (n, x)

[specfun package] return the Chebyshev function of the second kind for integers n > -1.

Reference: A&S, 22.8.3 page 783 and A&S 6.1.22 page 256.

To access this function, load("specfun").

See also CHEBYSHEV_T.

Function: GEN_LAGUERRE (n, a, x)

[specfun package] return the generalized Laguerre polynomial for integers n > -1.

To access this function, load("specfun").

Reference: table on page 789 in A&S.

Function: HERMITE (n,x)

[specfun package] return the Hermite polynomial for integers n > -1.

To access this function, load("specfun").

Reference: A&S 22.5.40 and 22.5.41, page 779.

Function: JACOBI_P (n, a, b, x)

[specfun package] return the Jacobi polynomial for integers n > -1 and a and b symbolic or a > -1 and b > -1. (The Jacobi polynomials are actually defined for all a and b ; however, the Jacobi polynomial weight (1-x)^a(1+x)^b isn't integrable for a <= -1 or b <= -1. )

When a, b, and x are floats (but not bfloats) specfun calls a special modedeclared version of jacobi_p. For numerical values, the modedeclared version is much faster than the other version. Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.

If n has been declared to be an integer, jacobi_p (n, a, b, x) returns a summation representation for the Jacobi function. Because Maxima simplifies 0^0 to 0 in a sum, two terms of the sum are added outside the summation.

To access this function, load("specfun").

Reference: table on page 789 in A&S.

Function: LAGUERRE (n, x)

[specfun package] return the Laguerre polynomial for integers n > -1.

Reference: A&S 22.5.16, page 778 and A&S page 789.

To access this function, load("specfun").

See also GEN_LAGUERRE.

Function: LEGENDRE_P (n, x)

[specfun package] return the Legendre polynomial of the first kind for integers n > -1.

Reference: A&S 22.5.35 page 779.

To access this function, load("specfun").

See LEGENDRE_Q.

Function: LEGENDRE_Q (n, x)

[specfun package] return the Legendre polynomial of the first kind for integers n > -1.

Reference: A&S 8.6.19 page 334.

To access this function, load("specfun").

See also LEGENDRE_P.

Function: SPHERICAL_BESSEL_J (n, x)

[specfun package] return the spherical Bessel function of the first kind for integers n > -1.

Reference: A&S 10.1.8 page 437 and A&S 10.1.15 page 439.

To access this function, load("specfun").

See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2, and SPHERICAL_BESSEL_Y.

Function: SPHERICAL_BESSEL_Y (n, x)

[specfun package] return the spherical Bessel function of the second kind for integers n > -1.

Reference: A&S 10.1.9 page 437 and 10.1.15 page 439.

To access this function, load("specfun").

See also SPHERICAL_HANKEL1, SPHERICAL_HANKEL2, and SPHERICAL_BESSEL_Y.

Function: SPHERICAL_HANKEL1 (n,x)

[specfun package] return the spherical hankel function of the first kind for integers n > -1.

Reference: A&S 10.1.36 page 439.

To access this function, load("specfun").

See also SPHERICAL_HANKEL2, SPHERICAL_BESSEL_J, and SPHERICAL_BESSEL_Y.

Function: SPHERICAL_HANKEL2 (n,x)

[specfun package] return the spherical hankel function of the second kind for integers n > -1.

Reference: A&S 10.1.17 page 439.

To access this function, load("specfun").

See also SPHERICAL_HANKEL1, SPHERICAL_BESSEL_J, and SPHERICAL_BESSEL_Y.

Function: SPHERICAL_HARMONIC (n, m, x, y)

[specfun package] return the spherical harmonic function for integers n > -1 and | m | <= n .

Reference: Merzbacher 9.64.

To access this function, load("specfun").

See also ASSOC_LEGENDRE_P

Function: ULTRASPHERICAL (n,a,x)

[specfun package] return the ultraspherical polynomials for integers n > -1. The ultraspherical polynomials are also known as Gegenbauer polynomials.

Reference: A&S 22.5.27

To access this function, load("specfun").

See also JACOBI_P.