Orthogonal polynomials¶

Chebyshev 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^{″} - 3 x 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) = 2 x T_{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) = 2 x, U_{n + 1} (x) = 2 x U_{n} (x) - U_{n - 1} (x) .

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

\begin{array}{r} \int_{- 1}^{1} T_{n} (x) T_{m} (x) \frac{d x}{\sqrt{1 - x^{2}}} = {\begin{array}{cl} 0 & if n \neq m, \\ π & if n = m = 0, \\ π / 2 & if n = m \neq 0, \end{array} \end{array}

and

\int_{- 1}^{1} U_{n} (x) U_{m} (x) \sqrt{1 - x^{2}} d x = \frac{π}{2} δ_{m, n} .

They are named after Pafnuty Chebyshev (1821-1894, alternative transliterations: Tchebyshef or Tschebyscheff).

Hermite polynomials¶

The Hermite polynomials are defined either by

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

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

H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{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}} d x = \sqrt{π} n! 2^{n} δ_{n m} .

They are named in honor of Charles Hermite (1822-1901), but were first introduced by Laplace in 1810 and also studied by Chebyshev in 1859.

Legendre polynomials¶

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}}{d x^{n}} [(x^{2} - 1)^{n}] .

These are solutions to Legendre’s differential equation:

\frac{d}{d x} [(1 - x^{2}) \frac{d}{d x} P (x)] + n (n + 1) P (x) = 0

and satisfy the orthogonality relation

\int_{- 1}^{1} P_{m} (x) P_{n} (x) d x = \frac{2}{2 n + 1} δ_{m n} .

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 $P_{ℓ}^{m} (x)$ can be given in terms of the “usual” Legendre polynomials by

\begin{array}{r} \begin{aligned} P_{ℓ}^{m} (x) & = (- 1)^{m} (1 - x^{2})^{m / 2} \frac{d^{m}}{d x^{m}} P_{ℓ} (x) \\ = \frac{(- 1)^{m}}{2^{ℓ} ℓ!} (1 - x^{2})^{m / 2} \frac{d^{ℓ + m}}{d x^{ℓ + m}} (x^{2} - 1)^{ℓ} . \end{aligned} \end{array}

Assuming $0 \leq m \leq ℓ$ , they satisfy the orthogonality relation:

\int_{- 1}^{1} P_{k}^{(m)} P_{ℓ}^{(m)} d x = \frac{2 (ℓ + m)!}{(2 ℓ + 1) (ℓ - m)!} δ_{k, ℓ},

where $δ_{k, ℓ}$ is the Kronecker delta.

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

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

They are named after Adrien-Marie Legendre (1752-1833).

Laguerre polynomials¶

Laguerre polynomials may be defined by the Rodrigues formula

L_{n} (x) = \frac{e^{x}}{n!} \frac{d^{n}}{d x^{n}} (e^{- x} x^{n}) .

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} d x = δ_{m n} .

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

L_{n}^{(α)} (x) = \frac{x^{- α} e^{x}}{n!} \frac{d^{n}}{d x^{n}} (e^{- x} x^{n + α}) .

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

They are named after Edmond Laguerre (1834-1886).

Jacobi polynomials¶

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}^{(α, β)} (z) = \frac{(α + 1)_{n}}{n!}_{2} F_{1} (- n, 1 + α + β + n; α + 1; \frac{1 - z}{2}),

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

P_{n}^{(α, β)} (z) = \frac{Γ (α + n + 1)}{n! Γ (α + β + n + 1)} \sum_{m = 0}^{n} (\binom{n}{m}) \frac{Γ (α + β + n + m + 1)}{Γ (α + m + 1)} {(\frac{z - 1}{2})}^{m} .

They are named after Carl Gustav Jaboc Jacobi (1804-1851).

Gegenbauer polynomials¶

Ultraspherical or Gegenbauer polynomials are given in terms of the Jacobi polynomials $P_{n}^{(α, β)} (x)$ with $α = β = a - 1 / 2$ by

C_{n}^{(a)} (x) = \frac{Γ (a + 1 / 2)}{Γ (2 a)} \frac{Γ (n + 2 a)}{Γ (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) d x = δ_{m n} 2^{1 - 2 a} π \frac{Γ (n + 2 a)}{(n + a) Γ^{2} (a) Γ (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{(2 a)^{\underset{―}{n}}}{n!}_{2} F_{1} (- n, 2 a + n; a + \frac{1}{2}; \frac{1 - z}{2})

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

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

Krawtchouk polynomials¶

The Krawtchouk polynomials are discrete orthogonal polynomials that are given by the hypergeometric series

K_{j} (x; n, p) = (- 1)^{j} (\binom{n}{j}) p^{j}_{2} F_{1} (- j, - x; - n; p^{- 1}) .

Since they are discrete orthogonal polynomials, they satisfy an orthogonality relation defined on a discrete (in this case finite) set of points:

\sum_{m = 0}^{n} K_{i} (m; n, p) K_{j} (m; n, p) (\binom{n}{m}) p^{m} q^{n - m} = (\binom{n}{j}) (p q)^{j} δ_{i j},

where $q = 1 - p$ . They can also be described by the recurrence relation

j K_{j} (x; n, p) = (x - (n - j + 1) p - (j - 1) q) K_{j - 1} (x; n, p) - p q (n - j + 2) K_{j - 2} (x; n, p),

where $K_{0} (x; n, p) = 1$ and $K_{1} (x; n, p) = x - n p$ .

They are named for Mykhailo Krawtchouk (1892-1942).

Meixner polynomials¶

The Meixner polynomials are discrete orthogonal polynomials that are given by the hypergeometric series

M_{n} (x; n, p) = (- 1)^{j} (\binom{n}{j}) p^{j}_{2} F_{1} (- j, - x; - n; p^{- 1}) .

They satisfy an orthogonality relation:

\sum_{k = 0}^{\infty} {\tilde{M}}_{n} (k; b, c) {\tilde{M}}_{m} (k; b, c) \frac{(b)_{k}}{k!} c^{k} = \frac{c^{- n} n!}{(b)_{n} (1 - c)^{b}} δ_{m n},

where ${\tilde{M}}_{n} (x; b, c) = M_{n} (x; b, c) / (b)_{x}$ , for $b > 0 ‘ a n d ‘ 0 < c < 1$ . They can also be described by the recurrence relation

\begin{array}{r} \begin{aligned} c (n - 1 + b) M_{n} (x; b, c) & = ((c - 1) x + n - 1 + c (n - 1 + b)) (b + n - 1) M_{n - 1} (x; b, c) \\ - (b + n - 1) (b + n - 2) (n - 1) M_{n - 2} (x; b, c), \end{aligned} \end{array}

where $M_{0} (x; b, c) = 0$ and $M_{1} (x; b, c) = (1 - c^{- 1}) x + b$ .

They are named for Josef Meixner (1908-1994).

Hahn polynomials¶

The Hahn polynomials are discrete orthogonal polynomials that are given by the hypergeometric series

Q_{k} (x; a, b, n) =_{3} F_{2} (- k, k + a + b + 1, - x; a + 1, - n; 1) .

They satisfy an orthogonality relation:

\sum_{k = 0}^{n - 1} Q_{i} (k; a, b, n) Q_{j} (k; a, b, n) ρ (k) = \frac{δ_{i j}}{π_{i}},

where

\begin{array}{r} \begin{aligned} ρ (k) & = (\binom{a + k}{k}) (\binom{b + n - k}{n - k}), \\ π_{i} & = δ_{i j} \frac{(- 1)^{i} i! (b + 1)_{i} (i + a + b + 1)_{n + 1}}{n! (2 i + a + b + 1) (- n)_{i} (a + 1)_{i}} . \end{aligned} \end{array}

They can also be described by the recurrence relation

A Q_{k} (x; a, b, n) = (- x + A + C) Q_{k - 1} (x; a, b, n) - C Q_{k - 2} (x; a, b, n),

where $Q_{0} (x; a, b, n) = 1$ and $Q_{1} (x; a, b, n) = 1 - \frac{a + b + 2}{(a + 1) n} x$ and

A = \frac{(k + a + b) (k + a) (n - k + 1)}{(2 k + a + b - 1) (2 k + a + b)}, C = \frac{(k - 1) (k + b - 1) (k + a + b + n)}{(2 k + a + b - 2) (2 k + a + b - 1)} .

They are named for Wolfgang Hahn (1911-1998), although they were first introduced by Chebyshev in 1875.

Pochhammer symbol¶

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) \dots (x + n - 1) = \frac{(x + n - 1)!}{(x - 1)!} .

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

x^{\underset{―}{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:

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=None)[source]¶

Bases: OrthogonalFunction

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

EXAMPLES:

sage: chebyshev_T(3, x)                                                         # needs sage.symbolic
4*x^3 - 3*x

class sage.functions.orthogonal_polys.Func_assoc_legendre_P[source]¶

Bases: BuiltinFunction

Return the Ferrers function $P_{n}^{m} (x)$ of first kind for $x \in (- 1, 1)$ with general order $m$ and general degree $n$ .

Ferrers functions of first kind are one of two linearly independent solutions of the associated Legendre differential equation

(1 - x^{2}) \frac{d^{2} w}{d x^{2}} - 2 x \frac{d w}{d x} + (n (n + 1) - \frac{m^{2}}{1 - x^{2}}) w = 0

on the interval $x \in (- 1, 1)$ and are usually denoted by $P_{n}^{m} (x)$ .

See also

The other linearly independent solution is called Ferrers function of second kind and denoted by $Q_{n}^{m} (x)$ , see Func_assoc_legendre_Q.

Warning

Ferrers functions must be carefully distinguished from associated Legendre functions which are defined on $C ∖ (- \infty, 1]$ and have not yet been implemented.

EXAMPLES:

We give the first Ferrers functions for nonnegative integers $n$ and $m$ in the interval $- 1 < x < 1$ :

sage: for n in range(4):                                                        # needs sage.symbolic
....:     for m in range(n+1):
....:         print(f"P_{n}^{m}({x}) = {gen_legendre_P(n, m, x)}")
P_0^0(x) = 1
P_1^0(x) = x
P_1^1(x) = -sqrt(-x^2 + 1)
P_2^0(x) = 3/2*x^2 - 1/2
P_2^1(x) = -3*sqrt(-x^2 + 1)*x
P_2^2(x) = -3*x^2 + 3
P_3^0(x) = 5/2*x^3 - 3/2*x
P_3^1(x) = -3/2*(5*x^2 - 1)*sqrt(-x^2 + 1)
P_3^2(x) = -15*(x^2 - 1)*x
P_3^3(x) = -15*(-x^2 + 1)^(3/2)

These expressions for nonnegative integers are computed by the Rodrigues-type given in eval_gen_poly(). Negative values for $n$ are obtained by the following identity:

P_{- n}^{m} (x) = P_{n - 1}^{m} (x) .

For $n$ being a nonnegative integer, negative values for $m$ are obtained by

P_{n}^{- | m |} (x) = (- 1)^{| m |} \frac{(n - | m |)!}{(n + | m |)!} P_{n}^{| m |} (x),

where $| m | \leq n$ .

Here are some specific values with negative integers:

sage: # needs sage.symbolic
sage: gen_legendre_P(-2, -1, x)
1/2*sqrt(-x^2 + 1)
sage: gen_legendre_P(2, -2, x)
-1/8*x^2 + 1/8
sage: gen_legendre_P(3, -2, x)
-1/8*(x^2 - 1)*x
sage: gen_legendre_P(1, -2, x)
0

Here are some other random values with floating numbers:

sage: # needs sage.symbolic
sage: m = var('m'); assume(m, 'integer')
sage: gen_legendre_P(m, m, .2)
0.960000000000000^(1/2*m)*(-1)^m*factorial(2*m)/(2^m*factorial(m))
sage: gen_legendre_P(.2, m, 0)
sqrt(pi)*2^m/(gamma(-1/2*m + 1.10000000000000)*gamma(-1/2*m + 0.400000000000000))
sage: gen_legendre_P(.2, .2, .2)
0.757714892929573

REFERENCES:

[DLMF-Legendre]

eval_gen_poly(n, m, arg, **kwds)[source]¶

Return the Ferrers function of first kind $P_{n}^{m} (x)$ for integers $n > - 1, m > - 1$ given by the following Rodrigues-type formula:

P_{n}^{m} (x) = (- 1)^{m + n} \frac{(1 - x^{2})^{m / 2}}{2^{n} n!} \frac{d^{m + n}}{d x^{m + n}} (1 - x^{2})^{n} .

INPUT:

n – integer degree
m – integer order
x – either an integer or a non-numerical symbolic expression

EXAMPLES:

sage: gen_legendre_P(7, 4, x)                                               # needs sage.symbolic
3465/2*(13*x^3 - 3*x)*(x^2 - 1)^2
sage: gen_legendre_P(3, 1, sqrt(x))                                         # needs sage.symbolic
-3/2*(5*x - 1)*sqrt(-x + 1)

REFERENCE:

[DLMF-Legendre], Section 14.7 eq. 10 (https://dlmf.nist.gov/14.7#E10)

class sage.functions.orthogonal_polys.Func_assoc_legendre_Q[source]¶

Bases: BuiltinFunction

EXAMPLES:

sage: loads(dumps(gen_legendre_Q))
gen_legendre_Q
sage: maxima(gen_legendre_Q(2, 1, 3, hold=True))._sage_().simplify_full()   # needs sage.symbolic
1/4*sqrt(2)*(36*pi - 36*I*log(2) + 25*I)

eval_recursive(n, m, x, **kwds)[source]¶

Return the associated Legendre Q(n, m, arg) function for integers $n > - 1, m > - 1$ .

EXAMPLES:

sage: # needs sage.symbolic
sage: gen_legendre_Q(3, 4, x)
48/(x^2 - 1)^2
sage: gen_legendre_Q(4, 5, x)
-384/((x^2 - 1)^2*sqrt(-x^2 + 1))
sage: gen_legendre_Q(0, 1, x)
-1/sqrt(-x^2 + 1)
sage: gen_legendre_Q(0, 2, x)
-1/2*((x + 1)^2 - (x - 1)^2)/(x^2 - 1)
sage: gen_legendre_Q(2, 2, x).subs(x=2).expand()
9/2*I*pi - 9/2*log(3) + 14/3

class sage.functions.orthogonal_polys.Func_chebyshev_T[source]¶

Bases: ChebyshevFunction

Chebyshev polynomials of the first kind.

REFERENCE:

[AS1964] 22.5.31 page 778 and 6.1.22 page 256.

EXAMPLES:

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

eval_algebraic(n, x)[source]¶

Evaluate chebyshev_T as polynomial, using a recursive formula.

INPUT:

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

EXAMPLES:

sage: chebyshev_T.eval_algebraic(5, x)                                      # needs sage.symbolic
2*(2*(2*x^2 - 1)*x - x)*(2*x^2 - 1) - x
sage: chebyshev_T(-7, x) - chebyshev_T(7, x)                                # needs sage.symbolic
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)                                               # needs sage.symbolic
sage: chebyshev_T(n, cos(x)).contains_zero()                                # needs sage.symbolic
True

sage: # needs sage.rings.padics
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)[source]¶

Evaluate chebyshev_T using an explicit formula. See [AS1964] 227 (p. 782) for details for the recursions. See also [Koe1999] for fast evaluation techniques.

INPUT:

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

EXAMPLES:

sage: # needs sage.symbolic
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(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

sage: chebyshev_T.eval_formula(2, 0.1) == chebyshev_T._evalf_(2, 0.1)       # needs sage.rings.complex_double
True

class sage.functions.orthogonal_polys.Func_chebyshev_U[source]¶

Bases: ChebyshevFunction

Class for the Chebyshev polynomial of the second kind.

REFERENCE:

[AS1964] 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)[source]¶

Evaluate chebyshev_U as polynomial, using a recursive formula.

INPUT:

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

EXAMPLES:

sage: chebyshev_U.eval_algebraic(5, x)                                      # needs sage.symbolic
-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)                                # needs sage.symbolic
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)                                                 # needs sage.symbolic
sage: chebyshev_U(n - 1, cos(x)).contains_zero()                            # needs sage.symbolic
True

sage: # needs sage.rings.padics
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)[source]¶

Evaluate chebyshev_U using an explicit formula.

See [AS1964] 227 (p. 782) for details on the recursions. See also [Koe1999] for the recursion formulas.

INPUT:

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

EXAMPLES:

sage: # needs sage.symbolic
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[source]¶

Bases: OrthogonalFunction

REFERENCE:

[AS1964] 22.5.16, page 778 and page 789.

class sage.functions.orthogonal_polys.Func_hahn[source]¶

Bases: OrthogonalFunction

Hahn polynomials $Q_{k} (x; a, b, n)$ .

INPUT:

k – the degree
x – the independent variable $x$
a, b – the parameters $a$ , $b$
n – the number of discrete points

EXAMPLES:

We verify the orthogonality for $n = 3$ :

sage: # needs sage.symbolic
sage: n = 2
sage: a, b = SR.var('a,b')
sage: def rho(k, a, b, n):
....:     return binomial(a + k, k) * binomial(b + n - k, n - k)
sage: M = matrix([[sum(rho(k, a, b, n)
....:                  * hahn(i, k, a, b, n) * hahn(j, k, a, b, n)
....:                  for k in range(n + 1)).expand().factor()
....:              for i in range(n+1)] for j in range(n+1)])
sage: M = M.factor()
sage: P = rising_factorial
sage: def diag(i, a, b, n):
....:    return ((-1)^i * factorial(i) * P(b + 1, i) * P(i + a + b + 1, n + 1)
....:            / (factorial(n) * (2*i + a + b + 1) * P(-n, i) * P(a + 1, i)))
sage: all(M[i,i] == diag(i, a, b, n) for i in range(3))
True
sage: all(M[i,j] == 0 for i in range(3) for j in range(3) if i != j)
True

eval_formula(k, x, a, b, n)[source]¶

Evaluate self using an explicit formula.

EXAMPLES:

sage: # needs sage.symbolic
sage: k, x, a, b, n = var('k,x,a,b,n')
sage: Q2 = hahn.eval_formula(2, x, a, b, n).simplify_full()
sage: Q2.coefficient(x^2).factor()
(a + b + 4)*(a + b + 3)/((a + 2)*(a + 1)*(n - 1)*n)
sage: Q2.coefficient(x).factor()
-(2*a*n - a + b + 4*n)*(a + b + 3)/((a + 2)*(a + 1)*(n - 1)*n)
sage: Q2(x=0)
1

eval_recursive(k, x, a, b, n, *args, **kwds)[source]¶

Return the Hahn polynomial $Q_{k} (x; a, b, n)$ using the recursive formula.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, a, b, n = var('x,a,b,n')
sage: hahn.eval_recursive(0, x, a, b, n)
1
sage: hahn.eval_recursive(1, x, a, b, n)
-(a + b + 2)*x/((a + 1)*n) + 1
sage: bool(hahn(2, x, a, b, n) == hahn.eval_recursive(2, x, a, b, n))
True
sage: bool(hahn(3, x, a, b, n) == hahn.eval_recursive(3, x, a, b, n))
True
sage: bool(hahn(4, x, a, b, n) == hahn.eval_recursive(4, x, a, b, n))
True
sage: M = matrix([[-1/2, -1], [1, 0]])                                      # needs sage.modules
sage: ret = hahn.eval_recursive(2, M, 1, 2, n).simplify_full().factor()     # needs sage.modules
sage: ret                                                                   # needs sage.modules
[1/4*(4*n^2 + 8*n - 19)/((n - 1)*n)          3/2*(4*n + 3)/((n - 1)*n)]
[        -3/2*(4*n + 3)/((n - 1)*n)          (n^2 - n - 7)/((n - 1)*n)]

class sage.functions.orthogonal_polys.Func_hermite[source]¶

Bases: GinacFunction

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

REFERENCE:

[AS1964] 22.5.40 and 22.5.41, page 779.

EXAMPLES:

sage: # needs sage.symbolic
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 Issue #17192 is fixed:

sage: # needs sage.symbolic
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
sage: m, x = SR.var('m,x')
sage: hermite(m, x).diff(m)
Traceback (most recent call last):
...
RuntimeError: derivative w.r.t. to the index is not supported yet

class sage.functions.orthogonal_polys.Func_jacobi_P[source]¶

Bases: OrthogonalFunction

Return 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 $(1 - x)^{a} (1 + x)^{b}$ is not integrable for $a \leq - 1$ or $b \leq - 1$ .

REFERENCE:

Table on page 789 in [AS1964].

EXAMPLES:

sage: x = PolynomialRing(QQ, 'x').gen()
sage: jacobi_P(2, 0, 0, x)                                                      # needs sage.libs.flint sage.symbolic
3/2*x^2 - 1/2
sage: jacobi_P(2, 1, 2, 1.2)                                                    # needs sage.libs.flint
5.01000000000000

class sage.functions.orthogonal_polys.Func_krawtchouk[source]¶

Bases: OrthogonalFunction

Krawtchouk polynomials $K_{j} (x; n, p)$ .

INPUT:

j – the degree
x – the independent variable $x$
n – the number of discrete points
p – the parameter $p$

See also

sage.coding.delsarte_bounds.krawtchouk() ${\bar{K}}_{l}^{n, q} (x)$ , which are related by

(- q)^{j} {\bar{K}}_{j}^{n, q^{- 1}} (x) = K_{j} (x; n, 1 - q) .

EXAMPLES:

We verify the orthogonality for $n = 4$ :

sage: n = 4
sage: p = SR.var('p')                                                           # needs sage.symbolic
sage: matrix([[sum(binomial(n,m) * p**m * (1-p)**(n-m)                          # needs sage.symbolic
....:              * krawtchouk(i,m,n,p) * krawtchouk(j,m,n,p)
....:              for m in range(n+1)).expand().factor()
....:          for i in range(n+1)] for j in range(n+1)])
[               1                0                0                0                0]
[               0     -4*(p - 1)*p                0                0                0]
[               0                0  6*(p - 1)^2*p^2                0                0]
[               0                0                0 -4*(p - 1)^3*p^3                0]
[               0                0                0                0    (p - 1)^4*p^4]

We verify the relationship between the Krawtchouk implementations:

sage: q = SR.var('q')                                                           # needs sage.symbolic
sage: all(codes.bounds.krawtchouk(n, 1/q, j, x)*(-q)^j                          # needs sage.symbolic
....:     == krawtchouk(j, x, n, 1-q) for j in range(n+1))
True

eval_formula(k, x, n, p)[source]¶

Evaluate self using an explicit formula.

EXAMPLES:

sage: x, n, p = var('x,n,p')                                                # needs sage.symbolic
sage: krawtchouk.eval_formula(3, x, n, p).expand().collect(x)               # needs sage.symbolic
-1/6*n^3*p^3 + 1/2*n^2*p^3 - 1/3*n*p^3 - 1/2*(n*p - 2*p + 1)*x^2
 + 1/6*x^3 + 1/6*(3*n^2*p^2 - 9*n*p^2 + 3*n*p + 6*p^2 - 6*p + 2)*x

eval_recursive(j, x, n, p, *args, **kwds)[source]¶

Return the Krawtchouk polynomial $K_{j} (x; n, p)$ using the recursive formula.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, n, p = var('x,n,p')
sage: krawtchouk.eval_recursive(0, x, n, p)
1
sage: krawtchouk.eval_recursive(1, x, n, p)
-n*p + x
sage: krawtchouk.eval_recursive(2, x, n, p).collect(x)
1/2*n^2*p^2 + 1/2*n*(p - 1)*p - n*p^2 + 1/2*n*p
 - 1/2*(2*n*p - 2*p + 1)*x + 1/2*x^2
sage: bool(krawtchouk.eval_recursive(2, x, n, p) == krawtchouk(2, x, n, p))
True
sage: bool(krawtchouk.eval_recursive(3, x, n, p) == krawtchouk(3, x, n, p))
True
sage: bool(krawtchouk.eval_recursive(4, x, n, p) == krawtchouk(4, x, n, p))
True

sage: M = matrix([[-1/2, -1], [1, 0]])                                      # needs sage.modules
sage: krawtchouk.eval_recursive(2, M, 3, 1/2)                               # needs sage.modules
[ 9/8  7/4]
[-7/4  1/4]

class sage.functions.orthogonal_polys.Func_laguerre[source]¶

Bases: OrthogonalFunction

REFERENCE:

[AS1964] 22.5.16, page 778 and page 789.

class sage.functions.orthogonal_polys.Func_legendre_P[source]¶

Bases: GinacFunction

EXAMPLES:

sage: # needs sage.symbolic
sage: legendre_P(4, 2.0)
55.3750000000000
sage: legendre_P(1, x)
x
sage: legendre_P(4, x + 1)
35/8*(x + 1)^4 - 15/4*(x + 1)^2 + 3/8
sage: legendre_P(1/2, I+1.)
1.05338240025858 + 0.359890322109665*I
sage: legendre_P(0, SR(1)).parent()
Symbolic Ring

sage: legendre_P(0, 0)                                                          # needs sage.symbolic
1
sage: legendre_P(1, x)                                                          # needs sage.symbolic
x

sage: # needs sage.symbolic
sage: legendre_P(4, 2.)
55.3750000000000
sage: legendre_P(5.5, 1.00001)
1.00017875754114
sage: legendre_P(1/2, I + 1).n()
1.05338240025858 + 0.359890322109665*I
sage: legendre_P(1/2, I + 1).n(59)
1.0533824002585801 + 0.35989032210966539*I
sage: legendre_P(42, RR(12345678))
2.66314881466753e309
sage: legendre_P(42, Reals(20)(12345678))
2.6632e309
sage: legendre_P(201/2, 0).n()
0.0561386178630179
sage: legendre_P(201/2, 0).n(100)
0.056138617863017877699963095883

sage: # needs sage.symbolic
sage: R.<x> = QQ[]
sage: legendre_P(4, x)
35/8*x^4 - 15/4*x^2 + 3/8
sage: legendre_P(10000, x).coefficient(x, 1)
0
sage: var('t,x')
(t, x)
sage: legendre_P(-5, t)
35/8*t^4 - 15/4*t^2 + 3/8
sage: legendre_P(4, x + 1)
35/8*(x + 1)^4 - 15/4*(x + 1)^2 + 3/8
sage: legendre_P(4, sqrt(2))
83/8
sage: legendre_P(4, I*e)
35/8*e^4 + 15/4*e^2 + 3/8

sage: # needs sage.symbolic
sage: n = var('n')
sage: derivative(legendre_P(n,x), x)
(n*x*legendre_P(n, x) - n*legendre_P(n - 1, x))/(x^2 - 1)
sage: derivative(legendre_P(3,x), x)
15/2*x^2 - 3/2
sage: derivative(legendre_P(n,x), n)
Traceback (most recent call last):
...
RuntimeError: derivative w.r.t. to the index is not supported yet

class sage.functions.orthogonal_polys.Func_legendre_Q[source]¶

Bases: BuiltinFunction

EXAMPLES:

sage: loads(dumps(legendre_Q))
legendre_Q
sage: maxima(legendre_Q(20, x, hold=True))._sage_().coefficient(x, 10)      # needs sage.symbolic
-29113619535/131072*log(-(x + 1)/(x - 1))

eval_formula(n, arg, **kwds)[source]¶

Return expanded Legendre Q(n, arg) function expression.

REFERENCE:

1. 1. Dunster, Legendre and Related Functions, https://dlmf.nist.gov/14.7#E2

EXAMPLES:

sage: # needs sage.symbolic
sage: legendre_Q.eval_formula(1, x)
1/2*x*(log(x + 1) - log(-x + 1)) - 1
sage: legendre_Q.eval_formula(2, x).expand().collect(log(1+x)).collect(log(1-x))
1/4*(3*x^2 - 1)*log(x + 1) - 1/4*(3*x^2 - 1)*log(-x + 1) - 3/2*x
sage: legendre_Q.eval_formula(20, x).coefficient(x, 10)
-29113619535/131072*log(x + 1) + 29113619535/131072*log(-x + 1)
sage: legendre_Q(0, 2)
-1/2*I*pi + 1/2*log(3)

sage: legendre_Q(0, 2.)                                                     # needs mpmath
0.549306144334055 - 1.57079632679490*I

eval_recursive(n, arg, **kwds)[source]¶

Return expanded Legendre Q(n, arg) function expression.

EXAMPLES:

sage: legendre_Q.eval_recursive(2, x)                                       # needs sage.symbolic
3/4*x^2*(log(x + 1) - log(-x + 1)) - 3/2*x - 1/4*log(x + 1) + 1/4*log(-x + 1)
sage: legendre_Q.eval_recursive(20, x).expand().coefficient(x, 10)          # needs sage.symbolic
-29113619535/131072*log(x + 1) + 29113619535/131072*log(-x + 1)

class sage.functions.orthogonal_polys.Func_meixner[source]¶

Bases: OrthogonalFunction

Meixner polynomials $M_{n} (x; b, c)$ .

INPUT:

n – the degree
x – the independent variable $x$
b, c – the parameters $b$ , $c$

eval_formula(n, x, b, c)[source]¶

Evaluate self using an explicit formula.

EXAMPLES:

sage: x, b, c = var('x,b,c')                                                # needs sage.symbolic
sage: meixner.eval_formula(3, x, b, c).expand().collect(x)                  # needs sage.symbolic
-x^3*(3/c - 3/c^2 + 1/c^3 - 1) + b^3
 + 3*(b - 2*b/c + b/c^2 - 1/c - 1/c^2 + 1/c^3 + 1)*x^2 + 3*b^2
 + (3*b^2 + 6*b - 3*b^2/c - 3*b/c - 3*b/c^2 - 2/c^3 + 2)*x + 2*b

eval_recursive(n, x, b, c, *args, **kwds)[source]¶

Return the Meixner polynomial $M_{n} (x; b, c)$ using the recursive formula.

EXAMPLES:

sage: # needs sage.symbolic
sage: x, b, c = var('x,b,c')
sage: meixner.eval_recursive(0, x, b, c)
1
sage: meixner.eval_recursive(1, x, b, c)
-x*(1/c - 1) + b
sage: meixner.eval_recursive(2, x, b, c).simplify_full().collect(x)
-x^2*(2/c - 1/c^2 - 1) + b^2 + (2*b - 2*b/c - 1/c^2 + 1)*x + b
sage: bool(meixner(2, x, b, c) == meixner.eval_recursive(2, x, b, c))
True
sage: bool(meixner(3, x, b, c) == meixner.eval_recursive(3, x, b, c))
True
sage: bool(meixner(4, x, b, c) == meixner.eval_recursive(4, x, b, c))
True
sage: M = matrix([[-1/2, -1], [1, 0]])
sage: ret = meixner.eval_recursive(2, M, b, c).simplify_full().factor()
sage: for i in range(2):  # make the output polynomials in 1/c
....:     for j in range(2):
....:         ret[i, j] = ret[i, j].collect(c)
sage: ret
[b^2 + 1/2*(2*b + 3)/c - 1/4/c^2 - 5/4    -2*b + (2*b - 1)/c + 3/2/c^2 - 1/2]
[    2*b - (2*b - 1)/c - 3/2/c^2 + 1/2             b^2 + b + 2/c - 1/c^2 - 1]

class sage.functions.orthogonal_polys.Func_ultraspherical[source]¶

Bases: GinacFunction

Return the ultraspherical (or Gegenbauer) polynomial gegenbauer(n,a,x),

C_{n}^{a} (x) = \sum_{k = 0}^{⌊ n / 2 ⌋} (- 1)^{k} \frac{Γ (n - k + a)}{Γ (a) k! (n - 2 k)!} (2 x)^{n - 2 k} .

When $n$ is a nonnegative integer, this formula gives a polynomial in $z$ of degree $n$ , but all parameters are permitted to be complex numbers. When $a = 1 / 2$ , the Gegenbauer polynomial reduces to a Legendre polynomial.

Computed using Pynac.

For numerical evaluation, consider using the mpmath library, as it also allows complex numbers (and negative $n$ as well); see the examples below.

REFERENCE:

[AS1964] 22.5.27

EXAMPLES:

sage: # needs sage.symbolic
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(1, 1, x)
2*x
sage: t = PolynomialRing(RationalField(), "t").gen()
sage: gegenbauer(3, 2, t)
32*t^3 - 12*t
sage: x = SR.var('x')
sage: n = ZZ.random_element(5, 5001)
sage: a = QQ.random_element().abs() + 5
sage: s = (  (n + 1)*ultraspherical(n + 1, a, x)
....:      - 2*x*(n + a)*ultraspherical(n, a, x)
....:      + (n + 2*a - 1)*ultraspherical(n - 1, a, x) )
sage: s.expand().is_zero()
True
sage: ultraspherical(5, 9/10, 3.1416)
6949.55439044240
sage: ultraspherical(5, 9/10, RealField(100)(pi))                               # needs sage.rings.real_mpfr
6949.4695419382702451843080687

sage: # needs sage.symbolic
sage: a, n = SR.var('a,n')
sage: gegenbauer(2, a, x)
2*(a + 1)*a*x^2 - a
sage: gegenbauer(3, a, x)
4/3*(a + 2)*(a + 1)*a*x^3 - 2*(a + 1)*a*x
sage: gegenbauer(3, a, x).expand()
4/3*a^3*x^3 + 4*a^2*x^3 + 8/3*a*x^3 - 2*a^2*x - 2*a*x
sage: gegenbauer(10, a, x).expand().coefficient(x, 2)
1/12*a^6 + 5/4*a^5 + 85/12*a^4 + 75/4*a^3 + 137/6*a^2 + 10*a
sage: ex = gegenbauer(100, a, x)
sage: (ex.subs(a==55/98) - gegenbauer(100, 55/98, x)).is_trivial_zero()
True

sage: # needs sage.symbolic
sage: gegenbauer(2, -3, x)
12*x^2 + 3
sage: gegenbauer(120, -99/2, 3)
1654502372608570682112687530178328494861923493372493824
sage: gegenbauer(5, 9/2, x)
21879/8*x^5 - 6435/4*x^3 + 1287/8*x
sage: gegenbauer(15, 3/2, 5)
3903412392243800

sage: derivative(gegenbauer(n, a, x), x)                                        # needs sage.symbolic
2*a*gegenbauer(n - 1, a + 1, x)
sage: derivative(gegenbauer(3, a, x), x)                                        # needs sage.symbolic
4*(a + 2)*(a + 1)*a*x^2 - 2*(a + 1)*a
sage: derivative(gegenbauer(n, a, x), a)                                        # needs sage.symbolic
Traceback (most recent call last):
...
RuntimeError: derivative w.r.t. to the second index is not supported yet

Numerical evaluation with the mpmath library:

sage: # needs mpmath
sage: from mpmath import gegenbauer as gegenbauer_mp
sage: from mpmath import mp
sage: print(gegenbauer_mp(-7,0.5,0.3))
0.1291811875
sage: with mp.workdps(25):
....:     print(gegenbauer_mp(2+3j, -0.75, -1000j))
(-5038991.358609026523401901 + 9414549.285447104177860806j)

class sage.functions.orthogonal_polys.OrthogonalFunction(name, nargs=2, latex_name=None, conversions=None)[source]¶

Bases: 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)[source]¶

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