Jacobi polynomials


In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

Definitions

Via the hypergeometric function

The Jacobi polynomials are defined via the hypergeometric function as follows:
where is Pochhammer's symbol. 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:
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 is the Gamma function.
In the special case that the four quantities, and 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.

Special cases

Basic properties

Orthogonality

The Jacobi polynomials satisfy the orthogonality condition
As defined, they do not have unit norm with respect to the weight. This can be corrected by dividing by the square root of the right hand side of the equation above, when.
Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:

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 is a solution of the second order linear homogeneous differential equation

Recurrence relations

The recurrence relation for the Jacobi polynomials of fixed α,β is:
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 = 1.

Asymptotics of Jacobi polynomials

For x in the interior of, the asymptotics of for large n is given by the Darboux formula
where
and the "O" term is uniform on the interval 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 is less explicit.

Applications

Wigner d-matrix

The expression allows the expression of the Wigner d-matrix djm’,m in terms of Jacobi polynomials: