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<title>Laguerre Functions - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">7.22 Laguerre Functions</h3>
<p><a name="index-Laguerre-functions-662"></a><a name="index-confluent-hypergeometric-function-663"></a>
The generalized Laguerre polynomials are defined in terms of confluent
hypergeometric functions as
<!-- {$L^a_n(x) = ((a+1)_n / n!) {}_1F_1(-n,a+1,x)$} -->
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 L_n(x) by L^0_n(x) = L_n(x) and
<!-- {$L^k_n(x) = (-1)^k (d^k/dx^k) L_{(n+k)}(x)$} -->
L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x). For
more information see Abramowitz & Stegun, Chapter 22.
<p>The functions described in this section are
declared in the header file <samp><span class="file">gsl_sf_laguerre.h</span></samp>.
<div class="defun">
— Function: double <b>gsl_sf_laguerre_1</b> (<var>double a, double x</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f1-664"></a></var><br>
— Function: double <b>gsl_sf_laguerre_2</b> (<var>double a, double x</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f2-665"></a></var><br>
— Function: double <b>gsl_sf_laguerre_3</b> (<var>double a, double x</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f3-666"></a></var><br>
— Function: int <b>gsl_sf_laguerre_1_e</b> (<var>double a, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f1_005fe-667"></a></var><br>
— Function: int <b>gsl_sf_laguerre_2_e</b> (<var>double a, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f2_005fe-668"></a></var><br>
— Function: int <b>gsl_sf_laguerre_3_e</b> (<var>double a, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flaguerre_005f3_005fe-669"></a></var><br>
<blockquote><p>These routines evaluate the generalized Laguerre polynomials
L^a_1(x), L^a_2(x), L^a_3(x) using explicit
representations.
<!-- Exceptional Return Values: none -->
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_sf_laguerre_n</b> (<var>const int n, const double a, const double x</var>)<var><a name="index-gsl_005fsf_005flaguerre_005fn-670"></a></var><br>
— Function: int <b>gsl_sf_laguerre_n_e</b> (<var>int n, double a, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005flaguerre_005fn_005fe-671"></a></var><br>
<blockquote><p>These routines evaluate the generalized Laguerre polynomials
L^a_n(x) for a > -1,
<!-- {$n \ge 0$} -->
n >= 0.
<!-- Domain: a > -1.0, n >= 0 -->
<!-- Evaluate generalized Laguerre polynomials. -->
<!-- Exceptional Return Values: GSL_EDOM -->
</blockquote></div>
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