Jacobi polynomials
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In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P(α, β)
n(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1 − x)α(1 + x)β on the interval [−1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.[1]
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
Contents
Definitions
Via the hypergeometric function
The Jacobi polynomials are defined via the hypergeometric function as follows:[2]
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:
Rodrigues' formula
An equivalent definition is given by Rodrigues' formula:[1][3]
If , then it reduces to the Legendre polynomials:
Alternate expression for real argument
For real x the Jacobi polynomial can alternatively be written as
and for integer n
where Γ(z) is the Gamma function.
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
-
(1)
The sum extends over all integer values of s for which the arguments of the factorials are nonnegative.
Basic properties
Orthogonality
The Jacobi polynomials satisfy the orthogonality condition
As defined, they are not orthonormal, the normalization being
Symmetry relation
The polynomials have the symmetry relation
thus the other terminal value is
Derivatives
The kth derivative of the explicit expression leads to
Differential equation
The Jacobi polynomial P(α, β)
n is a solution of the second order linear homogeneous differential equation[1]
Recurrence relations
The recurrence relation for the Jacobi polynomials of fixed α,β is:[1]
for n = 2, 3, ....
Since the Jacobi polynomials can be described in terms of the hypergeometric function, recurrences of the hypergeometric function give equivalent recurrences of the Jacobi polynomials. In particular, Gauss' contiguous relations correspond to the identities
Generating function
The generating function of the Jacobi polynomials is given by
where
and the branch of square root is chosen so that R(z, 0) = 1.[1]
Asymptotics of Jacobi polynomials
For x in the interior of [−1, 1], the asymptotics of P(α, β)
n for large n is given by the Darboux formula[1]
where
and the "O" term is uniform on the interval [ε, π-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.
Applications
Wigner d-matrix
The expression (1) allows the expression of the Wigner d-matrix djm’,m(φ) (for 0 ≤ φ ≤ 4π) in terms of Jacobi polynomials:[4]
See also
- Askey–Gasper inequality
- Big q-Jacobi polynomials
- Continuous q-Jacobi polynomials
- Little q-Jacobi polynomials
- Pseudo Jacobi polynomials
- Jacobi process
- Gegenbauer polynomials
- Romanovski polynomials
Notes
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Lua error in package.lua at line 80: module 'strict' not found. The definition is in IV.1; the differential equation – in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5.
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Further reading
- Lua error in package.lua at line 80: module 'strict' not found.
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