Jacobi polynomials

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

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight

$(1 - x)^\alpha (1+x)^\beta$

on the interval [-1, 1]. The Gegenbauer polynomials, and thus also the Legendre 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:

$P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!} \,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\frac{1-z}{2}\right) ,$

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:

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

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

$\begin{align} &P_n^{(\alpha,\beta)}(x)= (n+\alpha)! (n+\beta)! \\ &\qquad \times \sum_s \left[s! (n+\alpha-s)!(\beta+s)!(n-s)!\right]^{-1} \left(\frac{x-1}{2}\right)^{n-s} \left(\frac{x+1}{2}\right)^{s}. \end{align}$

(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

$\begin{align} &\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx \\ &\quad= \frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm} \end{align}$

for α > -1 and 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

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

Recurrent relation

The recurrent relation for the Jacobi polynomials is:

$\begin{align} &2n (n + \alpha + \beta) (2n + \alpha + \beta - 2) P_n^{(\alpha,\beta)}(z) \\ &\qquad= (2n+\alpha + \beta-1) \Big\{ (2n+\alpha + \beta)(2n+\alpha+\beta-2) z + \alpha^2 - \beta^2 \Big\} P_{n-1}^{(\alpha,\beta)}(z) \\ &\qquad\qquad - 2 (n+\alpha - 1) (n + \beta-1) (2n+\alpha + \beta) P_{n-2}^{(\alpha,\beta)}(z)~, \quad n = 2,3,\cdots \end{align}$

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

$R = R(z, w) = \big(1 - 2zw + w^2\big)^{1/2}~,$

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

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

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

where

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

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

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

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 ≤ 4π) in terms of Jacobi polynomials:

$\begin{align} &d^j_{m'm}(\phi) =\left[ \frac{(j+m)!(j-m)!}{(j+m')!(j-m')!}\right]^{1/2} \\ &\qquad\times \left(\sin\frac{\phi}{2}\right)^{m-m'} \left(\cos\frac{\phi}{2}\right)^{m+m'} P_{j-m}^{(m-m',m+m')}(\cos \phi). \end{align}$

See also

• Askey–Gasper inequality
• Big q-Jacobi polynomials
• Continuous q-Jacobi polynomials
• Little q-Jacobi polynomials
• Pseudo Jacobi polynomials
• Jacobi process

Notes

Further reading

• Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, ISBN 978-0-521-62321-6; 978-0-521-78988-2, MR1688958 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/18 

External links

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