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).

Implementation

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