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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.[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]

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

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:[1][3]

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

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[1]

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

Recurrence relation

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

\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.[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[1]

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

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:[4]

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

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.