Jacobi polynomials
Encyclopedia
In mathematics
, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials
. They are orthogonal with respect to the weight
on the interval [-1, 1]. The Gegenbauer polynomials
, and thus also the Legendre and Chebyshev polynomials
, are special cases of the Jacobi polynomials.
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
where
is Pochhammer's symbol
(for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:
:
written as
where s ≥ 0 and n-s ≥ 0, and for integer n
and Γ(z) is the Gamma function
, using the convention that:
In the special case that the four quantities
n, n+α, n+β, and
n+α+β are nonnegative integers,
the Jacobi polynomial can be written as
The sum extends over all integer values of s for which the arguments of the factorials are nonnegative.
for α > -1 and β > -1.
As defined, they are not orthonormal, the normalization being
thus the other terminal value is
of the Jacobi polynomials is given by
where
and the branch
of square root is chosen so that R(z, 0) = 0.
where
and the "O" term is uniform on the interval [ε, -ε] for every ε>0.
The asymptotics of the Jacobi polynomials near the points ±1 is given by the Mehler–Heine formula
where the limits are uniform for z in a bounded domain.
The asymptotics outside [-1, 1] is less explicit.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
. They are orthogonal with respect to the weight
on the interval [-1, 1]. The Gegenbauer polynomials
Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C are orthogonal polynomials on the interval [−1,1] with respect to the weight function α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials...
, and thus also the Legendre and Chebyshev polynomials
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
, are special cases of the Jacobi polynomials.
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
Via the hypergeometric function
The Jacobi polynomials are defined via the hypergeometric function as follows:where
is Pochhammer's symbolPochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
(for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:
Rodrigues' formula
An equivalent definition is given by Rodrigues' formulaRodrigues' formula
In mathematics, Rodrigues's formula is a formula for Legendre polynomials independently introduced by , and...
:
Alternate expression for real argument
For real x the Jacobi polynomial can alternatively bewritten as
where s ≥ 0 and n-s ≥ 0, and for integer n
and Γ(z) is the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
, using the convention that:
In the special case that the four quantities
n, n+α, n+β, and
n+α+β are nonnegative integers,
the Jacobi polynomial can be written as
The sum extends over all integer values of s for which the arguments of the factorials are nonnegative.
Orthogonality
The Jacobi polynomials satisfy the orthogonality conditionfor α > -1 and β > -1.
As defined, they are not orthonormal, the normalization being
Symmetry relation
The polynomials have the symmetry relationthus the other terminal value is
Derivatives
The kth derivative of the explicit expression leads toDifferential equation
The Jacobi polynomial Pn(α, β) is a solution of the second order linear homogeneous differential equationRecurrent relation
The recurrent relation for the Jacobi polynomials is:Generating function
The generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
of the Jacobi polynomials is given by
where
and the branch
Principal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....
of square root is chosen so that R(z, 0) = 0.
Asymptotics of Jacobi polynomials
For x in the interior of [-1, 1], the asymptotics of Pn(α,β) for large n is given by the Darboux formulawhere
and the "O" term is uniform on the interval [ε, -ε] for every ε>0.
The asymptotics of the Jacobi polynomials near the points ±1 is given by the Mehler–Heine formula
where the limits are uniform for z in a bounded domain.
The asymptotics outside [-1, 1] is less explicit.
Wigner d-matrix
The expression allows the expression of the Wigner d-matrix djm’,m(φ) (for 0 ≤ φ ≤ 4) in terms of Jacobi polynomials:See also
- Askey–Gasper inequalityAskey–Gasper inequalityIn mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture.-Statement:...
- Big q-Jacobi polynomialsBig q-Jacobi polynomialsIn mathematics, the big q-Jacobi polynomials Pn, introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme...
- Continuous q-Jacobi polynomialsContinuous q-Jacobi polynomialsIn mathematics, the continuous q-Jacobi polynomials P, introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.-Definition:...
- Little q-Jacobi polynomialsLittle q-Jacobi polynomialsIn mathematics, the little q-Jacobi polynomials pn are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by...
- Pseudo Jacobi polynomialsPseudo Jacobi polynomialsIn mathematics, the Pseudo Jacobi polynomials are a finite sequence Pn of orthogonal polynomials that can be expressed in terms of Jacobi polynomials of imaginary argument. They were introduced by ....
- Jacobi process