## 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 @math{C^{(\lambda)}_n(x)} using explicit representations for @math{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{$C^{(\lambda)}_n(x)$} @math{C^{(\lambda)}_n(x)} for a specific value of n, lambda, x subject to @math{\lambda > -1/2}, @c{$n \ge 0$} @math{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 @math{C^{(\lambda)}_n(x)} for @math{n = 0, 1, 2, \dots, nmax}, subject to @math{\lambda > -1/2}, @c{$nmax \ge 0$} @math{nmax >= 0}.