Orthogonal Polynomials¶

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.

Todo

Implement Zernike polynomials. Wikipedia article Zernike_polynomials

REFERENCES:

[ASHandbook]

(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) Abramowitz and Stegun: Handbook of Mathematical Functions, http://www.math.sfu.ca/ cbm/aands/

[EffCheby]

(1, 2) Wolfram Koepf: Effcient Computation of Chebyshev Polynomials in Computer Algebra Computer Algebra Systems: A Practical Guide. John Wiley, Chichester (1999): 79-99.

AUTHORS:

David Joyner (2006-06)
Stefan Reiterer (2010-)
Ralf Stephan (2015-)

The original module wrapped some of the orthogonal/special functions in the Maxima package “orthopoly” and was written by Barton Willis of the University of Nebraska at Kearney.

class sage.functions.orthogonal_polys.ChebyshevFunction(name, nargs=2, latex_name=None, conversions={})¶

Bases: sage.functions.orthogonal_polys.OrthogonalFunction

Abstract base class for Chebyshev polynomials of the first and second kind.

EXAMPLES:

sage: chebyshev_T(3,x)
4*x^3 - 3*x

class sage.functions.orthogonal_polys.Func_chebyshev_T¶

Bases: sage.functions.orthogonal_polys.ChebyshevFunction

Chebyshev polynomials of the first kind.

REFERENCE:

[ASHandbook] 22.5.31 page 778 and 6.1.22 page 256.

EXAMPLES:

sage: chebyshev_T(5,x)
16*x^5 - 20*x^3 + 5*x
sage: var('k')
k
sage: test = chebyshev_T(k,x)
sage: test
chebyshev_T(k, x)

eval_algebraic(n, x)¶

Evaluate chebyshev_T as polynomial, using a recursive formula.

INPUT:

n – an integer
x – a value to evaluate the polynomial at (this can be any ring element)

EXAMPLES:

sage: chebyshev_T.eval_algebraic(5, x)
2*(2*(2*x^2 - 1)*x - x)*(2*x^2 - 1) - x
sage: chebyshev_T(-7, x) - chebyshev_T(7,x)
0
sage: R.<t> = ZZ[]
sage: chebyshev_T.eval_algebraic(-1, t)
t
sage: chebyshev_T.eval_algebraic(0, t)
1
sage: chebyshev_T.eval_algebraic(1, t)
t
sage: chebyshev_T(7^100, 1/2)
1/2
sage: chebyshev_T(7^100, Mod(2,3))
2
sage: n = 97; x = RIF(pi/2/n)
sage: chebyshev_T(n, cos(x)).contains_zero()
True
sage: R.<t> = Zp(2, 8, 'capped-abs')[]
sage: chebyshev_T(10^6+1, t)
(2^7 + O(2^8))*t^5 + (O(2^8))*t^4 + (2^6 + O(2^8))*t^3 + (O(2^8))*t^2 + (1 + 2^6 + O(2^8))*t + (O(2^8))

eval_formula(n, x)¶

Evaluate chebyshev_T using an explicit formula. See [ASHandbook] 227 (p. 782) for details for the recurions. See also [EffCheby] for fast evaluation techniques.

INPUT:

n – an integer
x – a value to evaluate the polynomial at (this can be any ring element)

EXAMPLES:

sage: chebyshev_T.eval_formula(-1,x)
x
sage: chebyshev_T.eval_formula(0,x)
1
sage: chebyshev_T.eval_formula(1,x)
x
sage: chebyshev_T.eval_formula(2,0.1) == chebyshev_T._evalf_(2,0.1)
True
sage: chebyshev_T.eval_formula(10,x)
512*x^10 - 1280*x^8 + 1120*x^6 - 400*x^4 + 50*x^2 - 1
sage: chebyshev_T.eval_algebraic(10,x).expand()
512*x^10 - 1280*x^8 + 1120*x^6 - 400*x^4 + 50*x^2 - 1

class sage.functions.orthogonal_polys.Func_chebyshev_U¶

Bases: sage.functions.orthogonal_polys.ChebyshevFunction

Class for the Chebyshev polynomial of the second kind.

REFERENCE:

[ASHandbook] 22.8.3 page 783 and 6.1.22 page 256.

EXAMPLES:

sage: R.<t> = QQ[]
sage: chebyshev_U(2,t)
4*t^2 - 1
sage: chebyshev_U(3,t)
8*t^3 - 4*t

eval_algebraic(n, x)¶

Evaluate chebyshev_U as polynomial, using a recursive formula.

INPUT:

n – an integer
x – a value to evaluate the polynomial at (this can be any ring element)

EXAMPLES:

sage: chebyshev_U.eval_algebraic(5,x)
-2*((2*x + 1)*(2*x - 1)*x - 4*(2*x^2 - 1)*x)*(2*x + 1)*(2*x - 1)
sage: parent(chebyshev_U(3, Mod(8,9)))
Ring of integers modulo 9
sage: parent(chebyshev_U(3, Mod(1,9)))
Ring of integers modulo 9
sage: chebyshev_U(-3,x) + chebyshev_U(1,x)
0
sage: chebyshev_U(-1,Mod(5,8))
0
sage: parent(chebyshev_U(-1,Mod(5,8)))
Ring of integers modulo 8
sage: R.<t> = ZZ[]
sage: chebyshev_U.eval_algebraic(-2, t)
-1
sage: chebyshev_U.eval_algebraic(-1, t)
0
sage: chebyshev_U.eval_algebraic(0, t)
1
sage: chebyshev_U.eval_algebraic(1, t)
2*t
sage: n = 97; x = RIF(pi/n)
sage: chebyshev_U(n-1, cos(x)).contains_zero()
True
sage: R.<t> = Zp(2, 6, 'capped-abs')[]
sage: chebyshev_U(10^6+1, t)
(2 + O(2^6))*t + (O(2^6))

eval_formula(n, x)¶

Evaluate chebyshev_U using an explicit formula. See [ASHandbook] 227 (p. 782) for details on the recurions. See also [EffCheby] for the recursion formulas.

INPUT:

n – an integer
x – a value to evaluate the polynomial at (this can be any ring element)

EXAMPLES:

sage: chebyshev_U.eval_formula(10, x)
1024*x^10 - 2304*x^8 + 1792*x^6 - 560*x^4 + 60*x^2 - 1
sage: chebyshev_U.eval_formula(-2, x)
-1
sage: chebyshev_U.eval_formula(-1, x)
0
sage: chebyshev_U.eval_formula(0, x)
1
sage: chebyshev_U.eval_formula(1, x)
2*x
sage: chebyshev_U.eval_formula(2,0.1) == chebyshev_U._evalf_(2,0.1)
True

class sage.functions.orthogonal_polys.Func_gen_laguerre¶

Bases: sage.functions.orthogonal_polys.OrthogonalFunction

REFERENCE:

[ASHandbook] 22.5.16, page 778 and page 789.

class sage.functions.orthogonal_polys.Func_hermite¶

Bases: sage.symbolic.function.GinacFunction

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

REFERENCE:

[ASHandbook] 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)
64*w^3 - 24*w
sage: hermite(5,3.1416)
5208.69733891963
sage: hermite(5,RealField(100)(pi))
5208.6167627118104649470287166

Check that trac ticket #17192 is fixed:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: hermite(0,x)
1

sage: hermite(-1,x)
Traceback (most recent call last):
...
RuntimeError: hermite_eval: The index n must be a nonnegative integer

sage: hermite(-7,x)
Traceback (most recent call last):
...
RuntimeError: hermite_eval: The index n must be a nonnegative integer

class sage.functions.orthogonal_polys.Func_laguerre¶

Bases: sage.functions.orthogonal_polys.OrthogonalFunction

REFERENCE:

[ASHandbook] 22.5.16, page 778 and page 789.

class sage.functions.orthogonal_polys.Func_ultraspherical¶

Bases: sage.symbolic.function.GinacFunction

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

Computed using Maxima.

REFERENCE:

[ASHandbook] 22.5.27

EXAMPLES:

sage: ultraspherical(8, 101/11, x)
795972057547264/214358881*x^8 - 62604543852032/19487171*x^6...
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: var('x')
x
sage: for N in range(100):
....:     n = ZZ.random_element().abs() + 5
....:     a = QQ.random_element().abs() + 5
....:     assert ((n+1)*ultraspherical(n+1,a,x) - 2*x*(n+a)*ultraspherical(n,a,x) + (n+2*a-1)*ultraspherical(n-1,a,x)).expand().is_zero()
sage: ultraspherical(5,9/10,3.1416)
6949.55439044240
sage: ultraspherical(5,9/10,RealField(100)(pi))
6949.4695419382702451843080687

Check that trac ticket #17192 is fixed:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: ultraspherical(0,1,x)
1

sage: ultraspherical(-1,1,x)
Traceback (most recent call last):
...
RuntimeError: gegenb_eval: The index n must be a nonnegative integer

sage: ultraspherical(-7,1,x)
Traceback (most recent call last):
...
RuntimeError: gegenb_eval: The index n must be a nonnegative integer

class sage.functions.orthogonal_polys.OrthogonalFunction(name, nargs=2, latex_name=None, conversions={})¶

Bases: sage.symbolic.function.BuiltinFunction

Base class for orthogonal polynomials.

This class is an abstract base class for all orthogonal polynomials since they share similar properties. The evaluation as a polynomial is either done via maxima, or with pynac.

Convention: The first argument is always the order of the polynomial, the others are other values or parameters where the polynomial is evaluated.

eval_formula(*args)¶

Evaluate this polynomial using an explicit formula.

EXAMPLES:

sage: from sage.functions.orthogonal_polys import OrthogonalFunction
sage: P = OrthogonalFunction('testo_P')
sage: P.eval_formula(1,2.0)
Traceback (most recent call last):
...
NotImplementedError: no explicit calculation of values implemented

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

Returns the generalized (or associated) Legendre function of the first kind.

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*(5*t^2 - 1)*sqrt(-t^2 + 1)
sage: gen_legendre_P(4, 3, t)
105*(t^3 - t)*sqrt(-t^2 + 1)
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.

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.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 [ASHandbook].

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

Check that trac ticket #17192 is fixed:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: jacobi_P(0,0,0,x)
1

sage: jacobi_P(-1,0,0,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -1

sage: jacobi_P(-7,0,0,x)
Traceback (most recent call last):
...
ValueError: n must be greater than -1, got n = -7

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

Returns the Legendre polynomial of the first kind.

REFERENCE:

[ASHandbook] 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.

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

Orthogonal Polynomials¶

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