Jacobi polynomials

Via the hypergeometric function The Jacobi polynomials are defined via the hypergeometric function as follows: :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}(z) = \frac{1 }{2^n n! } \frac{d^n }{ d z^n } ( z^2 - 1 )^n \; . Differential equation The Jacobi polynomials P_n^{(\alpha,\beta)} is, up to scaling, the unique polynomial solution of the Sturm–Liouville problem : \left (1-x^2 \right )y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y' = \lambda y where \lambda = -n(n+\alpha+\beta+1). The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial solutions to Sturm–Liouville problems with polynomial coefficients. Alternate expression for real argument For real x the Jacobi polynomial can alternatively be written as :P_n^{(\alpha,\beta)}(x)= \sum_{s=0}^n {n+\alpha\choose n-s}{n+\beta \choose s} \left(\frac{x-1}{2}\right)^{s} \left(\frac{x+1}{2}\right)^{n-s} 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 where \Gamma(z) is the gamma function. In the special case that the four quantities n, n+\alpha, n+\beta, n+\alpha+\beta are nonnegative integers, the Jacobi polynomial can be written as {{NumBlk|:|P_n^{(\alpha,\beta)}(x)=(n+\alpha)! (n+\beta)! \sum_{s=0}^n \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}.|}} The sum extends over all integer values of s for which the arguments of the factorials are nonnegative. Special cases :P_0^{(\alpha,\beta)}(z)= 1, :P_1^{(\alpha,\beta)}(z)= (\alpha+1) + (\alpha+\beta+2)\frac{z-1}{2}, :P_2^{(\alpha,\beta)}(z)= \frac{(\alpha+1)(\alpha+2)}{2} + (\alpha+2)(\alpha+\beta+3)\frac{z-1}{2} + \frac{(\alpha+\beta+3)(\alpha+\beta+4)}{2}\left(\frac{z-1}{2}\right)^2. P_n^{(\alpha, \beta)}(z)=\frac{\Gamma(1 + 2 n+\alpha+\beta)}{\Gamma(1 + n)\Gamma(1 + n+\alpha+\beta)}\left(\frac{z}{2}\right)^n+\text { lower-degree terms }Thus, the leading coefficient is \frac{\Gamma(1 + 2 n+\alpha+\beta)}{2^nn! \Gamma(1 + n+\alpha+\beta)}. ==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 > -1. 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 n=m. Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity: :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{d^k}{dz^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). Recurrence relations The 3-term recurrence relation for the Jacobi polynomials of fixed \alpha, \beta is: : \begin{align} (z-1) \frac{d}{dz} P_n^{(\alpha,\beta)}(z) & = \frac{1}{2} (z-1)(1+\alpha+\beta+n)P_{n-1}^{(\alpha+1,\beta+1)} \\ & = n P_n^{(\alpha,\beta)} - (\alpha+n) P_{n-1}^{(\alpha,\beta+1)} \\ & =(1+\alpha+\beta+n) \left( P_n^{(\alpha,\beta+1)} - P_{n}^{(\alpha,\beta)} \right) \\ & =(\alpha+n) P_n^{(\alpha-1,\beta+1)} - \alpha P_n^{(\alpha,\beta)} \\ & =\frac{2(n+1) P_{n+1}^{(\alpha,\beta-1)} - \left(z(1+\alpha+\beta+n)+\alpha+1+n-\beta \right) P_n^{(\alpha,\beta)}}{1+z} \\ & =\frac{(2\beta+n+nz) P_n^{(\alpha,\beta)} - 2(\beta+n) P_n^{(\alpha,\beta-1)}}{1+z} \\ & =\frac{1-z}{1+z} \left( \beta P_n^{(\alpha,\beta)} - (\beta+n) P_{n}^{(\alpha+1,\beta-1)} \right) \, . \end{align} 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. Ultraspherical:\begin{aligned} C_n^{(\lambda)}(x) & =\frac{(2 \lambda)_n}{\left(\lambda+\frac{1}{2}\right)_n} P_n^{\left(\lambda-\frac{1}{2}, \lambda-\frac{1}{2}\right)}(x), \\ P_n^{(\alpha, \alpha)}(x) & =\frac{(\alpha+1)_n}{(2 \alpha+1)_n} C_n^{\left(\alpha+\frac{1}{2}\right)}(x) . \end{aligned}Legendre:P_n(x)=C_n^{\left(\frac{1}{2}\right)}(x)=P_n^{(0,0)}(x)Chebyshev:\begin{aligned} T_n(x)&=P_n^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x) / P_n^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1), \\ U_n(x)&=C_n^{(1)}(x)=(n+1) P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(x) / P_n^{\left(\frac{1}{2}, \frac{1}{2}\right)}(1), \\ V_n(x)&=P_n^{\left(-\frac{1}{2}, \frac{1}{2}\right)}(x) / P_n^{\left(-\frac{1}{2}, \frac{1}{2}\right)}(1), \\ W_n(x)&=(2 n+1) P_n^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x) / P_n^{\left(\frac{1}{2},-\frac{1}{2}\right)}(1) . \\ T_n^*(x)&=T_n(2 x-1), \\ U_n^*(x)&=U_n(2 x-1) . \end{aligned}Laguerre:\begin{aligned} \lim_{\beta \rightarrow \infty} P_n^{(\alpha, \beta)}(1-(2 x / \beta))&=L_n^{(\alpha)}(x) . \\ \lim_{\alpha \rightarrow \infty} P_n^{(\alpha, \beta)}((2 x / \alpha)-1)&=(-1)^n L_n^{(\beta)}(x) . \end{aligned}Hermite:\lim _{\alpha \rightarrow \infty} \alpha^{-\frac{1}{2} n} P_n^{(\alpha, \alpha)}\left(\alpha^{-\frac{1}{2}} x\right)=\frac{H_n(x)}{2^n n!} Stochastic process The Jacobi polynomials appear as the eigenfunctions of the Markov process on [-1, +1]\mathcal{L}=\left(1-x^2\right) \frac{\partial^2}{\partial x^2}+(p x+q) \frac{\partial}{\partial x} defined up to the time it hits the boundary. For p=-(\beta+\alpha+2), q=\beta-\alpha, we have\mathcal{L} P_n^{(\alpha, \beta)} = -n(n+\alpha+\beta+1) P_n^{(\alpha, \beta)}Thus this process is named the Jacobi process. Heat kernel Let • J^{(\alpha, \beta)} := -\left(1-x^2\right) \frac{d^2}{d x^2}-[\beta-\alpha-(\alpha+\beta+2) x] \frac{d}{d x} • T_t^{(\alpha, \beta)} := e^{-tJ^{(\alpha, \beta)}} • h_n^{(\alpha, \beta)}=\int_{-1}^1\left[P_n^{(\alpha, \beta)}(x)\right]^2(1-x)^\alpha(1+x)^\beta d x = \frac{2^{\alpha+\beta+1} \Gamma(n+\alpha+1) \Gamma(n+\beta+1)}{(2 n+\alpha+\beta+1) \Gamma(n+\alpha+\beta+1) \Gamma(n+1)} • G_t^{(\alpha, \beta)}(x, y)=\sum_{n=0}^{\infty} \exp (-t n(n+\alpha+\beta+1)) \frac{P_n^{(\alpha, \beta)}(x) P_n^{(\alpha, \beta)}(y)}{h_n^{(\alpha, \beta)}}, \quad x, y \in[-1,1], \quad t>0, • d \rho_{(\alpha, \beta)}(x)=(1-x)^\alpha(1+x)^\beta d x Then, for any f \in L^1\left(d \rho_{(\alpha, \beta)}\right),\operatorname{Disc}\left(P_n^{(\alpha, \beta)}\right)=2^{-n(n-1)} \prod_{j=1}^n j^{j-2 n+2}(j+\alpha)^{j-1}(j+\beta)^{j-1}(n+j+\alpha+\beta)^{n-j}Bailey’s formula:\begin{aligned} & \sum_{n=0}^{\infty} \frac{P_n^{(\alpha, \beta)}(\cos \theta) P_n^{(\alpha, \beta)}(\cos \varphi)}{h_n^{(\alpha, \beta)}} r^n=\frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1} \Gamma(\alpha+1) \Gamma(\beta+1)} \frac{1-r}{(1+r)^{\alpha+\beta+2}} \\ & \quad \times F_4\left(\frac{\alpha+\beta+2}{2}, \frac{\alpha+\beta+3}{2} ; \alpha+1, \beta+1 ;\left(\frac{2 \sin \frac{\theta}{2} \sin \frac{\varphi}{2}}{r^{1 / 2}+r^{-1 / 2}}\right)^2,\left(\frac{2 \cos \frac{\theta}{2} \cos \frac{\varphi}{2}}{r^{1 / 2}+r^{-1 / 2}}\right)^2\right) \end{aligned}where |r|-1, and F_4 is Appel's hypergeometric function of two variables. This is an analog of the Mehler kernel for Hermite polynomials, and the Hardy–Hille formula for Laguerre polynomials. Laplace-type integral representation:\begin{aligned} P_n^{\left(\alpha, \beta\right)}\left(1-2 t^2\right)= & \frac{(-1)^n 2^{2 n}}{\pi(2 n)!} \frac{\Gamma(n+\alpha+1) \Gamma(n+\beta+1)}{\Gamma\left(\alpha+\frac{1}{2}\right) \Gamma\left(\beta+\frac{1}{2}\right)} . \\ & \int_{-1}^1 \int_{-1}^1\left(t u \pm i \sqrt{1-t^2} v\right)^{2 n}\left(1-u^2\right)^{\alpha-\frac{1}{2}}\left(1-v^2\right)^{\beta-\frac{1}{2}} d u d v . \end{aligned} == Zeroes ==

If \alpha, \beta > -1, then P_n^{(\alpha, \beta)} has n real roots. Thus in this section we assume \alpha, \beta > -1 by default. This section is based on. Define: • j_{\alpha, m} are the positive zero of the Bessel function of the first kind J_\alpha, ordered such that 0 . • \theta_{n,m}=\theta_{n,m}^{(\alpha,\beta)} are the zeroes of P^{(\alpha,\beta)}_{n}\left(\cos\theta\right), ordered such that 0. • \rho=n+\frac{1}{2}(\alpha+\beta+1) • \phi_m=j_{\alpha, m} / \rho Inequalities \theta_{n, m} is strictly monotonically increasing with \alpha and strictly monotonically decreasing with \beta. \begin{aligned} \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{i} - x_{j}} &= \frac{1}{2}\left(\frac{\alpha+1}{1-x_i}-\frac{\beta+1}{1+x_i}\right)\\ \sum_{1 \leq j \leq n} \frac{1}{1 - x_{j}} &= \frac{n(n + \alpha + \beta + 1)}{2(\alpha + 1)} \\ \sum_{1 \leq j \leq n} \frac{1}{1 + x_{j}} &= \frac{n(n + \alpha + \beta + 1)}{2(\beta + 1)} \\ \sum_{1 \leq j \leq n} x_j &= \frac{n (\beta - \alpha)}{2n + \alpha + \beta} \end{aligned}The first relation can be interpreted physically. Fix an electric particle at +1 with charge \frac{1+\alpha}{2}, and another particle at -1 with charge \frac{1+\beta}{2}. Then, place n electric particles with charge +1 . The first relation states that the zeroes of P_n^{(\alpha, \beta)} are the equilibrium positions of the particles. This equilibrium is stable and unique. Other relations, such as \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^2}, \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^3}, are known in closed form. As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Jacobi polynomials. The electrostatic interpretation allows many relations to be intuitively seen. For example: • the symmetry relation between P_n^{(\alpha, \beta)} and P_n^{(\beta, \alpha)}; • the roots monotonically decrease when \alpha increases; Since the Stieltjes relation also exists for the Hermite polynomials and the Laguerre polynomials, by taking an appropriate limit of \alpha, \beta, the limit relations are derived. For example, for the Hermite polynomials, the zeros satisfy-x_{i} + \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{i}-x_{j}} = 0 Thus, by taking \alpha = \beta \to \infty limit, all the electric particles are forced into an infinitesimal neighborhood of the origin, where the field strength is linear. Then after scaling up the line, we obtain the same electrostatic configuration for the zeroes of Hermite polynomials. ==Asymptotics==

Darboux formula For x in the interior of [-1,1], the asymptotics of P_n^{(\alpha,\beta)} for large n is given by the Darboux formula • \mathrm{B} is the Euler beta function • (\cdot)_m is the falling factorial. • f_{m}(\theta)=\sum_{\ell=0}^{m}\frac{C_{m,\ell}(\alpha,\beta)}{\ell!(m-\ell)!}\frac{\cos\theta_{n,m,\ell}}{\left(\sin\frac{1}{2}\theta\right)^{\ell}\left(\cos\frac{1}{2}\theta\right)^{m-\ell}} • C_{m,\ell}(\alpha,\beta)={\left(\tfrac{1}{2}+\alpha\right)_{\ell}}{\left(\tfrac{1}{2}-\alpha\right)_{\ell}}{\left(\tfrac{1}{2}+\beta\right)_{m-\ell}}{\left(\tfrac{1}{2}-\beta\right)_{m-\ell}} • \theta_{n,m,\ell}=\tfrac{1}{2}(2n+\alpha+\beta+m+1)\theta-\tfrac{1}{2}(\alpha+\ell+\tfrac{1}{2})\pi Fix real \alpha, \beta, fix M = 1, 2, \dots, fix \delta \in (0, \pi/2). As n \to \infty,\left(\sin\tfrac{1}{2}\theta\right)^{\alpha+\frac{1}{2}}\left(\cos\tfrac{1}{2}\theta\right)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right) = {\pi}^{-1}2^{2n+\alpha+\beta+1}\mathrm{B}\left(n+\alpha+1,n+\beta+1\right)\left(\sum_{m=0}^{M-1}\frac{f_{m}(\theta)}{2^{m}{\left(2n+\alpha+\beta+2\right)_{m}}}+O\left(n^{-M}\right)\right) uniformly for all \theta \in [\delta , \pi - \delta]. The M = 1 case is the above Darboux formula. Hilb's type formula Define:(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos\tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)=\frac{\Gamma\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!} \left(\theta^{\frac{1}{2}}J_{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho\theta\right)\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right)where A_m, B_m are functions of \theta. The first few entries are:\begin{aligned} A_0(\theta) & =1 \\ \theta B_0(\theta) & =\frac{1}{4} g(\theta) \\ A_1(\theta) & =\frac{1}{8} g^{\prime}(\theta)-\frac{1+2 \alpha}{8} \frac{g(\theta)}{\theta}-\frac{1}{32}(g(\theta))^2 \end{aligned} For any fixed arbitrary constant c > 0, the error term satisfies\varepsilon_{M}(\rho,\theta)=\begin{cases}\theta O\left(\rho^{-2M-(3/2)}\right),&c\rho^{-1}\leq\theta\leq\pi-\delta,\\ \theta^{\alpha+(5/2)}O\left(\rho^{-2M+\alpha}\right),&0\leq\theta\leq c\rho^{-1},\end{cases} Mehler–Heine formula The asymptotics of the Jacobi polynomials near the points \pm 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 () allows the expression of the Wigner d-matrix d^j_{m',m}(\phi) (for 0\leq \phi\leq 4\pi) in terms of Jacobi polynomials: d^j_{m'm}(\phi) =(-1)^{\frac{m-m'-|m-m'|}{2}}\left[ \frac{(j+M)!(j-M)!}{(j+N)!(j-N)!}\right]^{\frac{1}{2}} \left(\sin\tfrac{\phi}{2}\right)^ \left(\cos\tfrac{\phi}{2}\right)^ P_{j-M}^{(|m-m'|,|m+m'|)}(\cos \phi), where M = \max(|m|,|m'|), N = \min(|m|,|m'|). ==See also==