| @cindex Laguerre functions |
| @cindex confluent hypergeometric function |
| |
| The generalized Laguerre polynomials are defined in terms of confluent |
| hypergeometric functions as |
| @c{$L^a_n(x) = ((a+1)_n / n!) {}_1F_1(-n,a+1,x)$} |
| @math{L^a_n(x) = ((a+1)_n / n!) 1F1(-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 |
| @c{$L^k_n(x) = (-1)^k (d^k/dx^k) L_{(n+k)}(x)$} |
| @math{L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x)}. For |
| more information see Abramowitz & Stegun, Chapter 22. |
| |
| The functions described in this section are |
| declared in the header file @file{gsl_sf_laguerre.h}. |
| |
| @deftypefun double gsl_sf_laguerre_1 (double @var{a}, double @var{x}) |
| @deftypefunx double gsl_sf_laguerre_2 (double @var{a}, double @var{x}) |
| @deftypefunx double gsl_sf_laguerre_3 (double @var{a}, double @var{x}) |
| @deftypefunx int gsl_sf_laguerre_1_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result}) |
| @deftypefunx int gsl_sf_laguerre_2_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result}) |
| @deftypefunx int gsl_sf_laguerre_3_e (double @var{a}, double @var{x}, gsl_sf_result * @var{result}) |
| These routines evaluate the generalized Laguerre polynomials |
| @math{L^a_1(x)}, @math{L^a_2(x)}, @math{L^a_3(x)} using explicit |
| representations. |
| @comment Exceptional Return Values: none |
| @end deftypefun |
| |
| |
| @deftypefun double gsl_sf_laguerre_n (const int @var{n}, const double @var{a}, const double @var{x}) |
| @deftypefunx int gsl_sf_laguerre_n_e (int @var{n}, double @var{a}, double @var{x}, gsl_sf_result * @var{result}) |
| These routines evaluate the generalized Laguerre polynomials |
| @math{L^a_n(x)} for @math{a > -1}, |
| @c{$n \ge 0$} |
| @math{n >= 0}. |
| |
| @comment Domain: a > -1.0, n >= 0 |
| @comment Evaluate generalized Laguerre polynomials. |
| @comment Exceptional Return Values: GSL_EDOM |
| @end deftypefun |
