Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials)

are a class of classical orthogonal polynomials.

They are orthogonal with respect to the weight

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.

The Jacobi polynomials are defined via the hypergeometric function as follows:[2] where

is Pochhammer's symbol (for the falling factorial).

In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression: An equivalent definition is given by Rodrigues' formula:[1][3] If

, then it reduces to the Legendre polynomials: For real

the Jacobi polynomial can alternatively be written as and for integer

is the gamma function.

In the special case that the four quantities

are nonnegative integers, the Jacobi polynomial can be written as The sum extends over all integer values of

for which the arguments of the factorials are nonnegative.

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: The polynomials have the symmetry relation thus the other terminal value is The

th derivative of the explicit expression leads to The Jacobi polynomial

is a solution of the second order linear homogeneous differential equation[1] The recurrence relation for the Jacobi polynomials of fixed

Writing for brevity

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 The generating function of the Jacobi polynomials is given by where and the branch of square root is chosen so that

is given by the Darboux formula[1] where and the "

" term is uniform on the interval

The asymptotics of the Jacobi polynomials near the points

is given by the Mehler–Heine formula where the limits are uniform for

in a bounded domain.

) in terms of Jacobi polynomials:[4]

( cos ⁡ ϕ ) ,

{\displaystyle d_{m'm}^{j}(\phi )=(-1)^{\frac {m-m'-|m-m'|}{2}}\left[{\frac {(j+M)!(j-M)!}{(j+N)!(j-N)!