Jacobi polynomials

Summary

In mathematics, Jacobi polynomials (occasionally called hypergeometric 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.[1]

Plot of the Jacobi polynomial function P n^(a,b) with n=10 and a=2 and b=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Jacobi polynomial function with and and in the complex plane from to with colors created with Mathematica 13.1 function ComplexPlot3D

The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

Definitions

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Via the hypergeometric function

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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:

 

Rodrigues' formula

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

 

If  , then it reduces to the Legendre polynomials:

 

Alternate expression for real argument

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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

  (1)

The sum extends over all integer values of   for which the arguments of the factorials are nonnegative.

Special cases

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Basic properties

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Orthogonality

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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

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The polynomials have the symmetry relation

 

thus the other terminal value is

 

Derivatives

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The  th derivative of the explicit expression leads to

 

Differential equation

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The Jacobi polynomial   is a solution of the second order linear homogeneous differential equation[1]

 

Recurrence relations

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The recurrence relation for the Jacobi polynomials of fixed  ,   is:[1]

 

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

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The generating function of the Jacobi polynomials is given by

 

where

 

and the branch of square root is chosen so that  .[1]


Asymptotics of Jacobi polynomials

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For   in the interior of  , the asymptotics of   for large   is given by the Darboux formula[1]

 

where

 

and the " " term is uniform on the interval   for every  .

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

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Wigner d-matrix

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The expression (1) allows the expression of the Wigner d-matrix   (for  ) in terms of Jacobi polynomials:[4]

 

where  .

See also

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Notes

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  1. ^ a b c d e f Szegő, Gábor (1939). "IV. Jacobi polynomials.". Orthogonal Polynomials. Colloquium Publications. Vol. 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; the asymptotic behavior is in VIII.2
  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. Vol. 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", Encyclopedia of Mathematics, EMS Press
  4. ^ Biedenharn, L.C.; Louck, J.D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley.

Further reading

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  • Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, ISBN 978-0-521-62321-6, MR 1688958, ISBN 978-0-521-78988-2
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
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