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:

Differential equation

The Jacobi polynomials is, up to scaling, the unique polynomial solution of the Sturm–Liouville problem
where. The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial solutions to Sturm–Liouville problems with polynomial coefficients.

Alternate expression for real argument

For real the Jacobi polynomial can alternatively be written as
and for integer
where 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.

Special cases

Thus, the leading coefficient is.

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 th derivative of the explicit expression leads to

Recurrence relations

The 3-term recurrence relation for the Jacobi polynomials of fixed, is:
for.
Writing for brevity, and, this becomes in terms of

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.

Other polynomials

The Jacobi polynomials reduce to other classical polynomials.
Ultraspherical:Legendre:Chebyshev:Laguerre:Hermite:

Stochastic process

The Jacobi polynomials appear as the eigenfunctions of the Markov process on defined up to the time it hits the boundary. For, we haveThus this process is named the Jacobi process.

Heat kernel

Let
Then, for any,Thus, is called the Jacobi heat kernel.

Other properties

The discriminant isBailey’s formula:where, and is Appel's hypergeometric function of two variables. This is an analog of the Mehler kernel for Hermite polynomials, and the Hardy–Hille formula for Laguerre polynomials.
Laplace-type integral representation:

Zeroes

If, then has real roots. Thus in this section we assume by default. This section is based on.
Define:

are the positive zero of the Bessel function of the first kind, ordered such that.
are the zeroes of, ordered such that.

Inequalities

is strictly monotonically increasing with and strictly monotonically decreasing with.
If, and, then is strictly monotonically increasing with.
When,

for
except when
for, except when
for

Asymptotics

Fix. Fix.
uniformly for.

Electrostatics

The zeroes satisfy the Stieltjes relations:
The first relation can be interpreted physically. Fix an electric particle at +1 with charge, and another particle at -1 with charge. Then, place electric particles with charge. The first relation states that the zeroes of are the equilibrium positions of the particles. This equilibrium is stable and unique.
Other relations, such as, are known in closed form.
As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Jacobi polynomials.
The electrostatic interpretation allows many relations to be intuitively seen. For example:

the symmetry relation between and ;
the roots monotonically decrease when increases;

Since the Stieltjes relation also exists for the Hermite polynomials and the Laguerre polynomials, by taking an appropriate limit of, the limit relations are derived. For example, for the Hermite polynomials, the zeros satisfyThus, by taking limit, all the electric particles are forced into an infinitesimal neighborhood of the origin, where the field strength is linear. Then after scaling up the line, we obtain the same electrostatic configuration for the zeroes of Hermite polynomials.

Asymptotics

Darboux formula

For in the interior of, the asymptotics of for large is given by the Darboux formula
where
and the "" term is uniform on the interval for every.
For higher orders, define:

is the Euler beta function
is the falling factorial.

Fix real, fix, fix. As,uniformly for all.
The case is the above Darboux formula.

Hilb's type formula

Define:

is the Bessel function

Fix real, fix. As, we have the Hilb's type formula:where are functions of. The first few entries are:
For any fixed arbitrary constant, the error term satisfies

Mehler–Heine formula

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.
The asymptotics outside is less explicit.

Applications

Wigner d-matrix

The expression allows the expression of the Wigner d-matrix

in terms of Jacobi polynomials:
where.