Jacobi polynomials

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

DefinitionsEdit

Via the hypergeometric functionEdit

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' formulaEdit

An equivalent definition is given by Rodrigues' formula:[1][3]

 

If  , then it reduces to the Legendre polynomials:

 

Alternate expression for real argumentEdit

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.

Special casesEdit

 
 
 

Basic propertiesEdit

OrthogonalityEdit

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 relationEdit

The polynomials have the symmetry relation

 

thus the other terminal value is

 

DerivativesEdit

The kth derivative of the explicit expression leads to

 

Differential equationEdit

The Jacobi polynomial P(α, β)
n
is a solution of the second order linear homogeneous differential equation[1]

 

Recurrence relationsEdit

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 functionEdit

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 polynomialsEdit

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.

ApplicationsEdit

Wigner d-matrixEdit

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 alsoEdit

NotesEdit

  1. ^ a b c d e f Szegő, Gábor (1939). "IV. Jacobi polynomials.". Orthogonal Polynomials. Colloquium Publications. XXIII. American Mathematical Society. ISBN 978-0-8218-1023-1. MR 0372517. 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.
  2. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 22". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 561. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  3. ^ P.K. Suetin (2001) [1994], "Jacobi_polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  4. ^ Biedenharn, L.C.; Louck, J.D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley.

Further readingEdit

External linksEdit