Jacobi polynomials

The Jacobi polynomials (after Carl Gustav Jacob Jacobi ), also hypergeometric polynomials are a set of polynomial solutions of the Sturm - Liouville problem, which form a set of orthogonal polynomials, on the interval with respect to the weighting function. You have the explicit form

Or by using the hypergeometric function 2F1:

Rodrigues formula

Recursion formulas

One can determine the Jacobi polynomials with the help of a recursion formula.

With the constants:

Properties

The value for

It applies the following symmetry relation

Resulting in the value of results:

They fulfill the orthogonality condition

Derivations

From the explicit form of the - th derivatives can be read. They are derived as:

Zeros

The eigenvalues of the symmetric tri-diagonal matrix

With

Coincide with the zeros of. Thus, the QR algorithm offers the possibility to calculate the zeros approximation. Furthermore, one can prove that they are simple and in the interval.

Asymptotic representation

With the help of the Landau symbols, the following formula can be set up:

Generating function

For all

The function

Is therefore referred to as a generating function of the Jacobi polynomials.

Special cases

Some important polynomials can be considered as special cases of the Jacobi polynomials:

For α = β = 0: Legendre polynomials
Gegenbauer polynomials
Chebyshev polynomials of first and second order
The centrifugal term of Zernike polynomials