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.

The Chebyshev polynomial of the first kind arises as a solution to the differential equation

$(1-x^2)\,y'' - x\,y' + n^2\,y = 0$

and those of the second kind as a solution to

$(1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0.$

The Chebyshev polynomials of the first kind are defined by the recurrence relation

$T_0(x) = 1 \, T_1(x) = x \, T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x). \,$

The Chebyshev polynomials of the second kind are defined by the recurrence relation

$U_0(x) = 1 \, U_1(x) = 2x \, U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). \,$

For integers $m,n$ , they satisfy the orthogonality relations

$\int_{-1}^1 T_n(x)T_m(x)\,\frac{dx}{\sqrt{1-x^2}} =\left\{ \begin{matrix} 0 &: n\ne m~~~~~\\ \pi &: n=m=0\\ \pi/2 &: n=m\ne 0 \end{matrix} \right.$

and

$\int_{-1}^1 U_n(x)U_m(x)\sqrt{1-x^2}\,dx =\frac{\pi}{2}\delta_{m,n}.$

They are named after Pafnuty Chebyshev (alternative transliterations: Tchebyshef or Tschebyscheff).
The Hermite polynomials are defined either by

$H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}$

(the “probabilists’ Hermite polynomials”), or by

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$

(the “physicists’ Hermite polynomials”). Sage (via Maxima) implements the latter flavor. These satisfy the orthogonality relation

$\int_{-\infty}^\infty H_n(x)H_m(x)\,e^{-x^2}\,dx ={n!2^n}{\sqrt{\pi}}\delta_{nm}$

They are named in honor of Charles Hermite.
Each Legendre polynomial $P_n(x)$ is an $n$ -th degree polynomial. It may be expressed using Rodrigues’ formula:

$P_n(x) = (2^n n!)^{-1} {\frac{d^n}{dx^n} } \left[ (x^2 -1)^n \right].$

These are solutions to Legendre’s differential equation:

${\frac{d}{dx}} \left[ (1-x^2) {\frac{d}{dx}} P(x) \right] + n(n+1)P(x) = 0.$

and satisfy the orthogonality relation

$\int_{-1}^{1} P_m(x) P_n(x)\,dx = {\frac{2}{2n + 1}} \delta_{mn}$

The Legendre function of the second kind $Q_n(x)$ 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 $P_\ell^m(x)$ can be given in terms of the “usual” Legendre polynomials by

$\begin{array}{ll} P_\ell^m(x) &= (-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}P_\ell(x) \\ &= \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell. \end{array}$

Assuming $0 \le m \le \ell$ , they satisfy the orthogonality relation:

$\int_{-1}^{1} P_k ^{(m)} P_\ell ^{(m)} dx = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell},$

where $\delta _{k,\ell}$ is the Kronecker delta.

The associated Legendre functions of the second kind $Q_\ell^m(x)$ can be given in terms of the “usual” Legendre polynomials by

$Q_\ell^m(x) = (-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}Q_\ell(x).$

They are named after Adrien-Marie Legendre.
Laguerre polynomials may be defined by the Rodrigues formula

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right).$

They are solutions of Laguerre’s equation:

$x\,y'' + (1 - x)\,y' + n\,y = 0\,$

and satisfy the orthogonality relation

$\int_0^\infty L_m(x) L_n(x) e^{-x}\,dx = \delta_{mn}.$

The generalized Laguerre polynomials may be defined by the Rodrigues formula:

$L_n^{(\alpha)}(x) = {\frac{x^{-\alpha} e^x}{n!}}{\frac{d^n}{dx^n}} \left(e^{-x} x^{n+\alpha}\right) .$

(These are also sometimes called the associated Laguerre polynomials.) The simple Laguerre polynomials are recovered from the generalized polynomials by setting $\alpha =0$ .

They are named after Edmond Laguerre.
Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite:

$P_n^{(\alpha,\beta)}(z) =\frac{(\alpha+1)_n}{n!} \,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\frac{1-z}{2}\right) ,$

where $()_n$ is Pochhammer’s symbol (for the rising factorial), (Abramowitz and Stegun p561.) and thus have the explicit expression

$P_n^{(\alpha,\beta)} (z) = \frac{\Gamma (\alpha+n+1)}{n!\Gamma (\alpha+\beta+n+1)} \sum_{m=0}^n {n\choose m} \frac{\Gamma (\alpha + \beta + n + m + 1)}{\Gamma (\alpha + m + 1)} \left(\frac{z-1}{2}\right)^m .$

They are named after Carl Jacobi.
Ultraspherical or Gegenbauer polynomials are given in terms of the Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ with $\alpha=\beta=a-1/2$ by

$C_n^{(a)}(x)= \frac{\Gamma(a+1/2)}{\Gamma(2a)}\frac{\Gamma(n+2a)}{\Gamma(n+a+1/2)} P_n^{(a-1/2,a-1/2)}(x).$

They satisfy the orthogonality relation

$\int_{-1}^1(1-x^2)^{a-1/2}C_m^{(a)}(x)C_n^{(a)}(x)\, dx =\delta_{mn}2^{1-2a}\pi \frac{\Gamma(n+2a)}{(n+a)\Gamma^2(a)\Gamma(n+1)} ,$

for $a>-1/2$ . They are obtained from hypergeometric series in cases where the series is in fact finite:

$C_n^{(a)}(z) =\frac{(2a)^{\underline{n}}}{n!} \,_2F_1\left(-n,2a+n;a+\frac{1}{2};\frac{1-z}{2}\right)$

where $\underline{n}$ is the falling factorial. (See Abramowitz and Stegun p561)

They are named for Leopold Gegenbauer (1849-1903).

For completeness, the Pochhammer symbol, introduced by Leo August Pochhammer, $(x)_n$ , is used in the theory of special functions to represent the “rising factorial” or “upper factorial”

$(x)_n=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}.$

On the other hand, the “falling factorial” or “lower factorial” is

$x^{\underline{n}}=\frac{x!}{(x-n)!} ,$

in the notation of Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics.

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

AUTHORS:

David Joyner (2006-06)

sage.functions.orthogonal_polys.chebyshev_T(n, x)¶

Returns the Chebyshev function of the first kind for integers $n>-1$ .

REFERENCE:

AS 22.5.31 page 778 and AS 6.1.22 page 256.

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: chebyshev_T(2,x)
2*x^2 - 1

sage.functions.orthogonal_polys.chebyshev_U(n, x)¶

Returns the Chebyshev function of the second kind for integers $n>-1$ .

REFERENCE:

AS, 22.8.3 page 783 and AS 6.1.22 page 256.

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: chebyshev_U(2,x)
4*x^2 - 1

sage.functions.orthogonal_polys.gegenbauer(n, a, x)¶

Returns the ultraspherical (or Gegenbauer) polynomial for integers $n > -1$ .

Computed using Maxima.

REFERENCE:

AS 22.5.27

EXAMPLES:

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

sage.functions.orthogonal_polys.gen_laguerre(n, a, x)¶

Returns the generalized Laguerre polynomial for integers $n > -1$ . Typically, a = 1/2 or a = -1/2.

REFERENCE:

table on page 789 in AS.

EXAMPLES:

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

sage.functions.orthogonal_polys.gen_legendre_P(n, m, x)¶

Returns the generalized (or associated) Legendre function of the first kind for integers $n > -1, m > -1$ .

The awkward code for when m is odd and 1 results from the fact that Maxima is happy with, for example, $(1 - t^2)^3/2$ , 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.

REFERENCE:

Gradshteyn and Ryzhik 8.706 page 1000.

EXAMPLES:

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*sqrt(-t^2 + 1)*(5*t^2 - 1)
sage: gen_legendre_P(4, 3, t) 
105*sqrt(-t^2 + 1)*(t^2 - 1)*t
sage: gen_legendre_P(7, 3, I).expand()
-16695*sqrt(2)
sage: gen_legendre_P(4, 1, 2.5)
-583.562373654533*I

sage.functions.orthogonal_polys.gen_legendre_Q(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.

EXAMPLES:

sage: P.<t> = QQ[]
sage: gen_legendre_Q(2,0,t)
3/4*t^2*log(-(t + 1)/(t - 1)) - 3/2*t - 1/4*log(-(t + 1)/(t - 1))
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(-x^2 + 1)
sage: gen_legendre_Q(2, 4, x).factor()
48*x/((x - 1)^2*(x + 1)^2)

sage.functions.orthogonal_polys.hermite(n, x)¶

Returns the Hermite polynomial for integers $n > -1$ .

REFERENCE:

AS 22.5.40 and 22.5.41, page 779.

EXAMPLES:

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)
8*(8*w^2 - 3)*w

sage.functions.orthogonal_polys.jacobi_P(n, a, b, x)¶

Returns the Jacobi polynomial $P_n^{(a,b)}(x)$ 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 $a \leq -1$ or $b \leq -1$ .

REFERENCE:

table on page 789 in AS.

EXAMPLES:

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

sage.functions.orthogonal_polys.laguerre(n, x)¶

Returns the Laguerre polynomial for integers $n > -1$ .

REFERENCE:

AS 22.5.16, page 778 and AS page 789.

EXAMPLES:

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

sage.functions.orthogonal_polys.legendre_P(n, x)¶

Returns the Legendre polynomial of the first kind for integers $n > -1$ .

REFERENCE:

AS 22.5.35 page 779.

EXAMPLES:

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)
7/2*I - 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

sage.functions.orthogonal_polys.legendre_Q(n, x)¶

Returns the Legendre function of the second kind for integers $n>-1$ .

Computed using Maxima.

EXAMPLES:

sage: P.<t> = QQ[]
sage: legendre_Q(2, t)
3/4*t^2*log(-(t + 1)/(t - 1)) - 3/2*t - 1/4*log(-(t + 1)/(t - 1))
sage: legendre_Q(3, 0.5)
-0.198654771479482
sage: legendre_Q(4, 2)
443/16*I*pi + 443/16*log(3) - 365/12
sage: legendre_Q(4, 2.0)
0.00116107583162324 + 86.9828465962674*I

sage.functions.orthogonal_polys.ultraspherical(n, a, x)¶

Returns the ultraspherical (or Gegenbauer) polynomial for integers $n > -1$ .

Computed using Maxima.

REFERENCE:

AS 22.5.27

EXAMPLES:

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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