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### Gegenbauer Polynomials

##### Synopsis

    #include <boost/math/special_functions/gegenbauer.hpp>

    namespace boost{ namespace math{

    template<typename Real>
    Real gegenbauer(unsigned n, Real lambda, Real x);

    template<typename Real>
    Real gegenbauer_prime(unsigned n, Real lambda, Real x);

    template<typename Real>
    Real gegenbauer_derivative(unsigned n, Real lambda, Real x, unsigned k);

    }} // namespaces

Gegenbauer polynomials are a family of orthogonal polynomials.

A basic usage is as follows:

    using boost::math::gegenbauer;
    double x = 0.5;
    double lambda = 0.5;
    unsigned n = 3;
    double y = gegenbauer(n, lambda, x);

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

    using boost::math::gegenbauer_derivative;
    double x = 0.5;
    double lambda = 0.5;
    unsigned n = 3;
    unsigned k = 2;
    double y = gegenbauer_derivative(n, lambda, x, k);

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

#### Implementation

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

#### Performance

Double precision timing on a consumer x86 laptop is shown below. Included is the time to generate a random number argument in the interval [-1, 1] (which takes 11.5ns).

    Run on (16 X 4300 MHz CPU s)
    CPU Caches:
      L1 Data 32K (x8)
      L1 Instruction 32K (x8)
      L2 Unified 1024K (x8)
      L3 Unified 11264K (x1)
    Load Average: 0.21, 0.33, 0.29
    -----------------------------------------
    Benchmark                            Time
    -----------------------------------------
    Gegenbauer<double>/1              12.5 ns
    Gegenbauer<double>/2              13.5 ns
    Gegenbauer<double>/3              14.6 ns
    Gegenbauer<double>/4              16.0 ns
    Gegenbauer<double>/5              17.5 ns
    Gegenbauer<double>/6              19.2 ns
    Gegenbauer<double>/7              20.7 ns
    Gegenbauer<double>/8              22.2 ns
    Gegenbauer<double>/9              23.6 ns
    Gegenbauer<double>/10             25.2 ns
    Gegenbauer<double>/11             26.9 ns
    Gegenbauer<double>/12             28.7 ns
    Gegenbauer<double>/13             30.5 ns
    Gegenbauer<double>/14             32.5 ns
    Gegenbauer<double>/15             34.3 ns
    Gegenbauer<double>/16             36.3 ns
    Gegenbauer<double>/17             38.0 ns
    Gegenbauer<double>/18             39.9 ns
    Gegenbauer<double>/19             41.8 ns
    Gegenbauer<double>/20             43.8 ns
    UniformReal<double>               11.5 ns

#### Accuracy

Some representative ULP plots are shown below. The relative accuracy cannot be controlled at the roots of the polynomial, as is to be expected.

#### Caveats

Some programs define the Gegenbauer polynomial with λ = 0 via renormalization (which makes them Chebyshev polynomials). We do not follow this convention: In this case, only the zeroth Gegenbauer polynomial is nonzero.