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<title>GNU Scientific Library &ndash; Reference Manual: Laguerre Functions</title>

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<a name="Laguerre-Functions"></a>
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<p>
Next: <a href="Lambert-W-Functions.html#Lambert-W-Functions" accesskey="n" rel="next">Lambert W Functions</a>, Previous: <a href="Hypergeometric-Functions.html#Hypergeometric-Functions" accesskey="p" rel="previous">Hypergeometric Functions</a>, Up: <a href="Special-Functions.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Laguerre-Functions-1"></a>
<h3 class="section">7.22 Laguerre Functions</h3>
<a name="index-Laguerre-functions"></a>
<a name="index-confluent-hypergeometric-function"></a>

<p>The generalized Laguerre polynomials are defined in terms of confluent
hypergeometric functions as
<em>L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x)</em>, and are sometimes referred to as the
associated Laguerre polynomials.  They are related to the plain
Laguerre polynomials <em>L_n(x)</em> by <em>L^0_n(x) = L_n(x)</em> and 
<em>L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x)</em>.  For
more information see Abramowitz &amp; Stegun, Chapter 22.
</p>
<p>The functions described in this section are
declared in the header file <samp>gsl_sf_laguerre.h</samp>.  
</p>
<dl>
<dt><a name="index-gsl_005fsf_005flaguerre_005f1"></a>Function: <em>double</em> <strong>gsl_sf_laguerre_1</strong> <em>(double <var>a</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005f2"></a>Function: <em>double</em> <strong>gsl_sf_laguerre_2</strong> <em>(double <var>a</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005f3"></a>Function: <em>double</em> <strong>gsl_sf_laguerre_3</strong> <em>(double <var>a</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005f1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_laguerre_1_e</strong> <em>(double <var>a</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005f2_005fe"></a>Function: <em>int</em> <strong>gsl_sf_laguerre_2_e</strong> <em>(double <var>a</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005f3_005fe"></a>Function: <em>int</em> <strong>gsl_sf_laguerre_3_e</strong> <em>(double <var>a</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines evaluate the generalized Laguerre polynomials
<em>L^a_1(x)</em>, <em>L^a_2(x)</em>, <em>L^a_3(x)</em> using explicit
representations.
</p></dd></dl>


<dl>
<dt><a name="index-gsl_005fsf_005flaguerre_005fn"></a>Function: <em>double</em> <strong>gsl_sf_laguerre_n</strong> <em>(const int <var>n</var>, const double <var>a</var>, const double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flaguerre_005fn_005fe"></a>Function: <em>int</em> <strong>gsl_sf_laguerre_n_e</strong> <em>(int <var>n</var>, double <var>a</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines evaluate the generalized Laguerre polynomials
<em>L^a_n(x)</em> for <em>a &gt; -1</em>, 
<em>n &gt;= 0</em>.
</p>
</dd></dl>




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