### 7.20 Gegenbauer Functions

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials. The functions described in this section are declared in the header file `gsl_sf_gegenbauer.h`.

Function: double gsl_sf_gegenpoly_1 (double lambda, double x)
Function: double gsl_sf_gegenpoly_2 (double lambda, double x)
Function: double gsl_sf_gegenpoly_3 (double lambda, double x)
Function: int gsl_sf_gegenpoly_1_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_2_e (double lambda, double x, gsl_sf_result * result)
Function: int gsl_sf_gegenpoly_3_e (double lambda, double x, gsl_sf_result * result)

  These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.

Function: double gsl_sf_gegenpoly_n (int n, double lambda, double x)
Function: int gsl_sf_gegenpoly_n_e (int n, double lambda, double x, gsl_sf_result * result)

  These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, n >= 0.

Function: int gsl_sf_gegenpoly_array (int nmax, double lambda, double x, double result_array[])

  This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0.