

.. _laguerre-functions:

Laguerre Functions
==================

The generalized Laguerre polynomials are defined in terms of confluent
hypergeometric functions as :math:`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 :math:`L_n(x)` by
:math:`L^0_n(x) = L_n(x)` and :math:`L^k_n(x) = (-1)^k (d^k/dx^k) L_{n+k}(x)`.
For more information see Abramowitz & Stegun, Chapter 22.

.. index::
  Laguerre function
  Laguerre polynomial

.. function:: gsl_sf_laguerre_1(a, x)

.. function:: gsl_sf_laguerre_2(a, x)

.. function:: gsl_sf_laguerre_3(a, x)

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

.. function:: gsl_sf_laguerre_n(n, a, x)

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