Jacobi polynomials

From Wikipedia, the free encyclopedia

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

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^{(α, β)}(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

Via the hypergeometric function

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

$P_{n}^{{(\alpha ,\beta )}}(z)={\frac {(\alpha +1)_{n}}{n!}}\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\frac {1-z}{2}}\right),$

where $(\alpha +1)_{n}$ 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:

$P_{n}^{{(\alpha ,\beta )}}(z)={\frac {\Gamma (\alpha +n+1)}{n!\,\Gamma (\alpha +\beta +n+1)}}\sum _{{m=0}}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m}~.$

Rodrigues' formula

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

$P_{n}^{{(\alpha ,\beta )}}(z)={\frac {(-1)^{n}}{2^{n}n!}}(1-z)^{{-\alpha }}(1+z)^{{-\beta }}{\frac {d^{n}}{dz^{n}}}\left\{(1-z)^{\alpha }(1+z)^{\beta }(1-z^{2})^{n}\right\}~.$

Alternate expression for real argument

For real x the Jacobi polynomial can alternatively be written as

$P_{n}^{{(\alpha ,\beta )}}(x)=\sum _{s}{n+\alpha \choose s}{n+\beta \choose n-s}\left({\frac {x-1}{2}}\right)^{{n-s}}\left({\frac {x+1}{2}}\right)^{{s}}$

where s ≥ 0 and n−s ≥ 0, and for integer n

${z \choose n}={\frac {\Gamma (z+1)}{\Gamma (n+1)\Gamma (z-n+1)}},$

and Γ(z) is the Gamma function, using the convention that:

${z \choose n}=0\quad {\text{for}}\quad n<0.$

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

$P_{n}^{{(\alpha ,\beta )}}(x)=(n+\alpha )!(n+\beta )!\sum _{s}{\frac {1}{s!(n+\alpha -s)!(\beta +s)!(n-s)!}}\left({\frac {x-1}{2}}\right)^{{n-s}}\left({\frac {x+1}{2}}\right)^{{s}}.$

(1)

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

Basic properties

Orthogonality

The Jacobi polynomials satisfy the orthogonality condition

$\int _{{-1}}^{1}(1-x)^{{\alpha }}(1+x)^{{\beta }}P_{m}^{{(\alpha ,\beta )}}(x)P_{n}^{{(\alpha ,\beta )}}(x)\;dx={\frac {2^{{\alpha +\beta +1}}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{{nm}}$

for α, β > −1.

As defined, they are not orthonormal, the normalization being

$P_{n}^{{(\alpha ,\beta )}}(1)={n+\alpha \choose n}.$

Symmetry relation

The polynomials have the symmetry relation

$P_{n}^{{(\alpha ,\beta )}}(-z)=(-1)^{n}P_{n}^{{(\beta ,\alpha )}}(z);$

thus the other terminal value is

$P_{n}^{{(\alpha ,\beta )}}(-1)=(-1)^{n}{n+\beta \choose n}.$

Derivatives

The kth derivative of the explicit expression leads to

${\frac {{\mathrm d}^{k}}{{\mathrm d}z^{k}}}P_{n}^{{(\alpha ,\beta )}}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)}}P_{{n-k}}^{{(\alpha +k,\beta +k)}}(z).$

Differential equation

The Jacobi polynomial P_n^{(α, β)} is a solution of the second order linear homogeneous differential equation^[1]

$(1-x^{2})y''+(\beta -\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.$

Recurrence relation

The recurrent relation for the Jacobi polynomials is:^[1]

${\begin{aligned}&2n(n+\alpha +\beta )(2n+\alpha +\beta -2)P_{n}^{{(\alpha ,\beta )}}(z)=\\&\quad =(2n+\alpha +\beta -1){\Big \{}(2n+\alpha +\beta )(2n+\alpha +\beta -2)z+\alpha ^{2}-\beta ^{2}{\Big \}}P_{{n-1}}^{{(\alpha ,\beta )}}(z)-2(n+\alpha -1)(n+\beta -1)(2n+\alpha +\beta )P_{{n-2}}^{{(\alpha ,\beta )}}(z)~,\end{aligned}}$

for n = 2, 3, ....

Generating function

The generating function of the Jacobi polynomials is given by

$\sum _{{n=0}}^{\infty }P_{n}^{{(\alpha ,\beta )}}(z)w^{n}=2^{{\alpha +\beta }}R^{{-1}}(1-w+R)^{{-\alpha }}(1+w+R)^{{-\beta }}~,$

where

$R=R(z,w)=\left(1-2zw+w^{2}\right)^{{{\frac {1}{2}}}}~,$

and the branch of square root is chosen so that R(z, 0) = 1.^[1]

Asymptotics of Jacobi polynomials

For x in the interior of [-1, 1], the asymptotics of P_n^{(α, β)} for large n is given by the Darboux formula^[1]

$P_{n}^{{(\alpha ,\beta )}}(\cos \theta )=n^{{-{\frac {1}{2}}}}k(\theta )\cos(N\theta +\gamma )+O(n^{{-{\frac {3}{2}}}})~,$

where

${\begin{aligned}k(\theta )&=\pi ^{{-{\frac {1}{2}}}}\sin ^{{-\alpha -{\frac {1}{2}}}}{\frac {\theta }{2}}\cos ^{{-\beta -{\frac {1}{2}}}}{\frac {\theta }{2}}~,\\N&=n+{\frac {\alpha +\beta +1}{2}}~,\\\gamma &=-\left(\alpha +{\tfrac {1}{2}}\right){\frac {\pi }{2}}~,\end{aligned}}$

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

${\begin{aligned}\lim _{{n\to \infty }}n^{{-\alpha }}P_{n}^{{(\alpha ,\beta )}}\left(\cos {\frac {z}{n}}\right)&=\left({\frac {z}{2}}\right)^{{-\alpha }}J_{\alpha }(z)~,\\\lim _{{n\to \infty }}n^{{-\beta }}P_{n}^{{(\alpha ,\beta )}}\left(\cos \left[\pi -{\frac {z}{n}}\right]\right)&=\left({\frac {z}{2}}\right)^{{-\beta }}J_{\beta }(z)~,\end{aligned}}$

where the limits are uniform for z in a bounded domain.

The asymptotics outside [−1, 1] is less explicit.

Applications

Wigner d-matrix

The expression (1) allows the expression of the Wigner d-matrix d^j_m’,m(φ) (for 0 ≤ φ ≤ 4 $π$ ) in terms of Jacobi polynomials:^[4]

$d_{{m'm}}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{{{\frac {1}{2}}}}\left(\sin {\frac {\phi }{2}}\right)^{{m-m'}}\left(\cos {\frac {\phi }{2}}\right)^{{m+m'}}P_{{j-m}}^{{(m-m',m+m')}}(\cos \phi ).$

Notes

↑ 1.0 1.1 1.2 1.3 1.4 1.5 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.
↑ Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 22", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 561, ISBN 978-0486612720, MR 0167642 .
↑ P.K. Suetin (2001), "Jacobi_polynomials", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
↑ Biedenharn, L.C.; Louck, J.D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley.

External links

Weisstein, Eric W., "Jacobi Polynomial", MathWorld.

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.