Gegenbauer Functions

The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter 22, where they are known as Ultraspherical polynomials.

gsl_sf_gegenpoly_1(lambda, x)
gsl_sf_gegenpoly_2(lambda, x)
gsl_sf_gegenpoly_3(lambda, x)

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

gsl_sf_gegenpoly_n(n, lambda, x)

This function evaluates the Gegenbauer polynomial \(C^{(\lambda)}_n(x)\) for a specific value of \(n\), \(\lambda\), \(x\) subject to \(\lambda > -1/2\), \(n \geq 0\).