Module: sage.functions.orthogonal_polys
Orthogonal Polynomials
This module wraps some of the orthogonal/special functions in the Maxima package "orthopoly". This package was written by Barton Willis of the University of Nebraska at Kearney. It is released under the terms of the General Public License (GPL). Send Maxima-related bug reports and comments on this module to willisb@unk.edu. In your report, please include Maxima and specfun version information.
and those of the second kind as a solution to
The Chebyshev polynomials of the first kind are defined by the recurrence relation
The Chebyshev polynomials of the second kind are defined by the recurrence relation
For integers
, they satisfy the orthogonality relations
and
They are named after Pafnuty Chebyshev (alternative transliterations: Tchebyshef or Tschebyscheff).
(the "probabilists' Hermite polynomials"), or by
(the "physicists' Hermite polynomials"). SAGE (via Maxima) implements the latter flavor. These satisfy the orthogonality relation
They are named in honor of Charles Hermite.
These are solutions to Legendre's differential equation:
and satisfy the orthogonality relation
The Legendre function of the second kind
is another
(linearly independent) solution to the Legendre differential equation.
It is not an ``orthogonal polynomial'' however.
The associated Legendre functions of the first kind
can be given in terms of the ``usual'' Legendre polynomials by
Assuming
where
The associated Legendre functions of the second kind
can be given in terms of the ``usual'' Legendre polynomials by
They are named after Adrien-Marie Legendre.
They are solutions of Laguerre's equation:
and satisfy the orthogonality relation
The generalized Laguerre polynomials may be defined by the Rodrigues formula:
(These are also sometimes called the associated Laguerre polynomials.) The simple Laguerre polynomials are recovered from the generalized polynomials by setting
They are named after Edmond Laguerre.
where
They are named after Carl Jacobi.
They satisfy the orthogonality relation
for
where
They are named for Leopold Gegenbauer (1849-1903).
For completeness, the Pochhammer symbol, introduced by
Leo August Pochhammer,
, is used in the theory of special
functions to represent the "rising factorial" or "upper factorial"
On the other hand, the "falling factorial" or "lower factorial" is
in the notation of Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics.
Methods implemented: * chebyshev_T (n, x) - the Chebyshev polynomial of the first kind for integers n > -1. REFERENCE: A&S 22.5.31 page 778 and A&S 6.1.22 page 256. * chebyshev_U (n, x) - the Chebyshev polynomial of the second kind for integers n > -1. REFERENCE: A&S, 22.8.3 page 783 and A&S 6.1.22 page 256. * gen_laguerre (n, a, x) - the generalized Laguerre polynomial for integers n > -1. REFERENCE: table on page 789 in A&S. * gen_legendre_P (n, x) - the associated (or generalized) Legendre function of the first kind for integers n > -1. * gen_legendre_Q (n, x) - the associated (or generalized) Legendre function of the second kind for integers n > -1. REFERENCE: Gradshteyn and Ryzhik 8.706 page 1000. * hermite (n,x) - the Hermite polynomial for integers n > -1. REFERENCE: A&S 22.5.40 and 22.5.41, page 779. * jacobi_P (n, a, b, x) - the Jacobi polynomial for integers n > -1 and a and b symbolic or a > -1 and b > -1. REFERENCE: table on page 789 in A&S. * laguerre (n, x) - the generalized Laguerre polynomial for integers n > -1. REFERENCE: A&S 22.5.16, page 778 and A&S page 789. * legendre_P (n, x) - the Legendre polynomial of the first kind for integers n > -1. REFERENCE: A&S 22.5.35 page 779. * legendre_Q (n, x) - the Legendre function of the second kind for integers n > -1. * ultraspherical (n,a,x) - the ultraspherical polynomials for integers n > -1. The ultraspherical polynomials are also known as Gegenbauer polynomials. REFERENCE: A&S 22.5.27
NOTE: The first call of any of these will usually cost a bit extra (it loads "specfun", but I'm not sure if that is the real reason). The next call is usually faster but not always.
TODO: Implement associated Legendre polynomials and Zernike polynomials. (Neither is in Maxima.) http://en.wikipedia.org/wiki/Associated_Legendre_polynomials http://en.wikipedia.org/wiki/Zernike_polynomials
REFERENCES: * Abramowitz and Stegun: Handbook of Mathematical Functions, http://www.math.sfu.ca/ cbm/aands/ * http://en.wikipedia.org/wiki/Chebyshev_polynomials * http://en.wikipedia.org/wiki/Legendre_polynomials * http://en.wikipedia.org/wiki/Hermite_polynomials * http://mathworld.wolfram.com/GegenbauerPolynomial.html * http://en.wikipedia.org/wiki/Jacobi_polynomials * http://en.wikipedia.org/wiki/Laguerre_polynomia * http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
Author: David Joyner (2006-06)
Module-level Functions
n, x) |
Returns the Chebyshev function of the first kind for integers n > -1.
sage: x = PolynomialRing(QQ, 'x').gen() sage: chebyshev_T(2,x) 2*x^2 - 1
n, x) |
Returns the Chebyshev function of the second kind for integers n > -1.
sage: x = PolynomialRing(QQ, 'x').gen() sage: chebyshev_U(2,x) 4*x^2 - 1
n, a, x) |
Returns the ultraspherical (or Gegenbauer) polynomial for integers n > -1.
Computed using Maxima.
sage: x = PolynomialRing(QQ, 'x').gen() sage: ultraspherical(2,3/2,x) 15/2*x^2 - 3/2 sage: ultraspherical(2,1/2,x) 3/2*x^2 - 1/2 sage: ultraspherical(1,1,x) 2*x sage: t = PolynomialRing(RationalField(),"t").gen() sage: gegenbauer(3,2,t) 32*t^3 - 12*t
n, a, x) |
Returns the generalized Laguerre polynomial for integers n > -1. Typically, a = 1/2 or a = -1/2.
sage: x = PolynomialRing(QQ, 'x').gen() sage: gen_laguerre(2,1,x) 1/2*x^2 - 3*x + 3 sage: gen_laguerre(2,1/2,x) 1/2*x^2 - 5/2*x + 15/8 sage: gen_laguerre(2,-1/2,x) 1/2*x^2 - 3/2*x + 3/8 sage: gen_laguerre(2,0,x) 1/2*x^2 - 2*x + 1 sage: gen_laguerre(3,0,x) -1/6*x^3 + 3/2*x^2 - 3*x + 1
n, m, x) |
Returns the generalized (or associated) Legendre function of the
first kind for integers
.
The awkward code for when m is odd and > 1 results from the fact that Maxima is
happy with, for example,
, but SAGE is not. For these cases the
function is computed from the (m-1)-case using one of the recursions satisfied
by the Legendre functions.
sage: P.<t> = QQ[] sage: gen_legendre_P(2, 0, t) 3/2*t^2 - 1/2 sage: gen_legendre_P(2, 0, t) == legendre_P(2, t) True sage: gen_legendre_P(3, 1, t) (3/2 - 15*t^2/2)*sqrt(1 - t^2) sage: gen_legendre_P(4, 3, t) sqrt(1 - t^2)*(105*t^3 - 105*t) sage: gen_legendre_P(7, 3, I).expand() -16695*sqrt(2) sage: gen_legendre_P(4, 1, 2.5) -583.562373654533*I
n, m, x) |
Returns the generalized (or associated) Legendre function of the second kind for integers n > -1, m > -1.
Maxima restricts m <= n. Hence the cases m > n are computed using the same recursion used for gen_legendre_P(n,m,x) when m is odd and > 1.
sage: P.<t> = QQ[] sage: gen_legendre_Q(2,0,t) (3*log((-t - 1)/(t - 1))*t^2 - 6*t - log((-t - 1)/(t - 1)))/4 sage: gen_legendre_Q(2,0,t) - legendre_Q(2, t) 0 sage: gen_legendre_Q(3,1,0.5) 2.49185259170895 sage: gen_legendre_Q(0, 1, x) -1/sqrt(1 - x^2) sage: gen_legendre_Q(2, 4, x).factor() 48*x/((x - 1)^2*(x + 1)^2)
n, x) |
Returns the Hermite polynomial for integers
.
sage: x = PolynomialRing(QQ, 'x').gen() sage: hermite(2,x) 4*x^2 - 2 sage: hermite(3,x) 8*x^3 - 12*x sage: hermite(3,2) 40 sage: S.<y> = PolynomialRing(RR) sage: hermite(3,y) 8.00000000000000*y^3 - 12.0000000000000*y sage: R.<x,y> = QQ[] sage: hermite(3,y^2) 8*y^6 - 12*y^2 sage: w = var('w') sage: hermite(3,2*w) -24*w*(1 - 8*w^2/3)
n, a, b, x) |
Returns the Jacobi polynomial
for integers
and a and b symbolic or
and
.
The Jacobi polynomials are actually defined for all a and b.
However, the Jacobi polynomial weight
isn't
integrable for
or
.
sage: x = PolynomialRing(QQ, 'x').gen() sage: jacobi_P(2,0,0,x) 3/2*x^2 - 1/2 sage: jacobi_P(2,1,2,1.2) # random output of low order bits 5.009999999999998
n, x) |
Returns the Laguerre polynomial for integers
.
sage: x = PolynomialRing(QQ, 'x').gen() sage: laguerre(2,x) 1/2*x^2 - 2*x + 1 sage: laguerre(3,x) -1/6*x^3 + 3/2*x^2 - 3*x + 1 sage: laguerre(2,2) -1
n, x) |
Returns the Legendre polynomial of the first kind for integers n > -1.
sage: P.<t> = QQ[] sage: legendre_P(2,t) 3/2*t^2 - 1/2 sage: legendre_P(3, 1.1) 1.67750000000000 sage: legendre_P(3, 1 + I) 7*I/2 - 13/2 sage: legendre_P(3, MatrixSpace(ZZ, 2)([1, 2, -4, 7])) [-179 242] [-484 547] sage: legendre_P(3, GF(11)(5)) 8
n, x) |
Returns the Legendre function of the second kind for integers n > -1.
Computed using Maxima.
sage: P.<t> = QQ[] sage: legendre_Q(2, t) (3*log((-t - 1)/(t - 1))*t^2 - 6*t - log((-t - 1)/(t - 1)))/4 sage: legendre_Q(3, 0.5) -0.198654771479482 sage: legendre_Q(4, 2) (1329*log(-3) - 1460)/48 sage: legendre_Q(4, 2.0) NaN
n, a, x) |
Returns the ultraspherical (or Gegenbauer) polynomial for integers n > -1.
Computed using Maxima.
sage: x = PolynomialRing(QQ, 'x').gen() sage: ultraspherical(2,3/2,x) 15/2*x^2 - 3/2 sage: ultraspherical(2,1/2,x) 3/2*x^2 - 1/2 sage: ultraspherical(1,1,x) 2*x sage: t = PolynomialRing(RationalField(),"t").gen() sage: gegenbauer(3,2,t) 32*t^3 - 12*t
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