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In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) Pn(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.

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:

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

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:

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

If $$\alpha = \beta = 0 , then it reduces to the Legendre polynomials: \( P_{n} = \frac{1 }{2^n n! } \frac{d^n }{ d z^n } ( z^2 - 1 )^n \; .$$

Alternate expression for real argument

For real x the Jacobi polynomial can alternatively be written as

$$P_n^{(\alpha,\beta)}(x)= \sum_s {n+\alpha\choose s}{n+\beta \choose n-s} \left(\frac{x-1}{2}\right)^{n-s} \left(\frac{x+1}{2}\right)^{s}, \qquad n \geq s \geq 0.$$

and for integer n

$${z \choose n} = \begin{cases} \frac{\Gamma(z+1)}{\Gamma(n+1)\Gamma(z-n+1)} & n \geq 0 \\ 0 & n < 0 \end{cases}$$

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

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

$$\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx =\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm}, \qquad \alpha, \beta, \alpha+\beta > -1.$$

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(α, β)
n is a solution of the second order linear homogeneous differential equation

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

Recurrence relation

The recurrence 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) - 2 (n+\alpha - 1) (n + \beta-1) (2n+\alpha + \beta) P_{n-2}^{(\alpha, \beta)}(z), \end{align}

for n = 2, 3, ....
Generating function

The generating function of the Jacobi polynomials is given by

$$\sum_{n=0}^\infty P_n^{(\alpha,\beta)}(z) t^n = 2^{\alpha + \beta} R^{-1} (1 - t + R)^{-\alpha} (1 + t + R)^{-\beta},$$

where

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

and the branch of square root is chosen so that R(z, 0) = 1.
Asymptotics of Jacobi polynomials

For x in the interior of [−1, 1], the asymptotics of Pn(α, β) (x) for large n is given by the Darboux formula

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

where

\begin{align} k(\theta) &= \pi^{-\frac{1}{2}} \sin^{-\alpha-\frac{1}{2}} \tfrac{\theta}{2} \cos^{-\beta-\frac{1}{2}} \tfrac{\theta}{2},\\ N &= n + \tfrac{1}{2} (\alpha+\beta+1),\\ \gamma &= - \tfrac{\pi}{2} \left (\alpha + \tfrac{1}{2} \right ), \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 \left ( \tfrac{z}{n} \right ) \right) &= \left(\tfrac{z}{2}\right)^{-\alpha} J_\alpha(z)\\ \lim_{n \to \infty} n^{-\beta}P_n^{(\alpha,\beta)}\left(\cos \left (\pi - \tfrac{z}{n} \right) \right) &= \left(\tfrac{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 djm’,m(φ) (for 0 ≤ φ ≤ 4π) in terms of Jacobi polynomials:

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

Big q-Jacobi polynomials
Continuous q-Jacobi polynomials
Little q-Jacobi polynomials
Pseudo Jacobi polynomials
Jacobi process
Gegenbauer polynomials

Notes

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.