Laguerre Functions

The generalized Laguerre polynomials are defined in terms of confluent hypergeometric functions as \(L^a_n(x) = ((a+1)_n / n!) {}_1F_1(-n,a+1,x)\), and are sometimes referred to as the associated Laguerre polynomials. They are related to the plain Laguerre polynomials \(L_n(x)\) by \(L^0_n(x) = L_n(x)\) and \(L^k_n(x) = (-1)^k (d^k/dx^k) L_{n+k}(x)\). For more information see Abramowitz & Stegun, Chapter 22.

gsl_sf_laguerre_1(a, x)
gsl_sf_laguerre_2(a, x)
gsl_sf_laguerre_3(a, x)

These routines evaluate the generalized Laguerre polynomials \(L^a_1(x), L^a_2(x), L^a_3(x)\) using explicit representations.

gsl_sf_laguerre_n(n, a, x)

This routine evaluates the generalized Laguerre polynomials \(L^a_n(x)\) for \(a > -1, n >= 0\).