Jacobi polynomials

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For Jacobi polynomials of several variables, see Heckman–Opdam 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.

Definitions[edit]

Via the hypergeometric function[edit]

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' formula[edit]

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

If , then it reduces to the Legendre polynomials:

Alternate expression for real argument[edit]

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.

Basic properties[edit]

Orthogonality[edit]

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[edit]

The polynomials have the symmetry relation

thus the other terminal value is

Derivatives[edit]

The kth derivative of the explicit expression leads to

Differential equation[edit]

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

Recurrence relations[edit]

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 function[edit]

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 polynomials[edit]

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.

Applications[edit]

Wigner d-matrix[edit]

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 also[edit]

Notes[edit]

  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), "Jacobi_polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  4. ^ Biedenharn, L.C.; Louck, J.D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. 

Further reading[edit]

External links[edit]