[/
  Copyright 2019, Nick Thompson
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

[section:jacobi Jacobi Polynomials]

[h4 Synopsis]

``
#include <boost/math/special_functions/jacobi.hpp>
``

   namespace boost{ namespace math{

   template<typename Real>
   Real jacobi(unsigned n, Real alpha, Real beta, Real x);

   template<typename Real>
   Real jacobi_derivative(unsigned n, Real alpha, Real beta, Real x, unsigned k);

   template<typename Real>
   Real jacobi_prime(unsigned n, Real alpha, Real beta, Real x);

   template<typename Real>
   Real jacobi_double_prime(unsigned n, Real alpha, Real beta, Real x);

   }} // namespaces

Jacobi polynomials are a family of orthogonal polynomials.

A basic usage is as follows:

    using boost::math::jacobi;
    double x = 0.5;
    double alpha = 0.3;
    double beta = 7.2;
    unsigned n = 3;
    double y = jacobi(n, alpha, beta, x);

All derivatives of the Jacobi polynomials are available.
The /k/-th derivative of the /n/-th Gegenbauer polynomial is given by

    using boost::math::jacobi_derivative;
    double x = 0.5;
    double alpha = 0.3;
    double beta = 7.2;
    unsigned n = 3;
    double y = jacobi_derivative(n, alpha, beta, x, k);

For consistency with the rest of the library, `jacobi_prime` is provided which simply returns `jacobi_derivative(n, lambda, x,1)`.

[$../graphs/jacobi.svg]

[h3 Implementation]

The implementation uses the 3-term recurrence for the Jacobi polynomials, rising.


[endsect]