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<a name="Gegenbauer-Functions"></a>
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<p>
Next: <a href="Hypergeometric-Functions.html#Hypergeometric-Functions" accesskey="n" rel="next">Hypergeometric Functions</a>, Previous: <a href="Gamma-and-Beta-Functions.html#Gamma-and-Beta-Functions" accesskey="p" rel="previous">Gamma and Beta 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="Gegenbauer-Functions-1"></a>
<h3 class="section">7.20 Gegenbauer Functions</h3>
<a name="index-Gegenbauer-functions"></a>

<p>The Gegenbauer polynomials are defined in Abramowitz &amp; Stegun, Chapter
22, where they are known as Ultraspherical polynomials.  The functions
described in this section are declared in the header file
<samp>gsl_sf_gegenbauer.h</samp>.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f1"></a>Function: <em>double</em> <strong>gsl_sf_gegenpoly_1</strong> <em>(double <var>lambda</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f2"></a>Function: <em>double</em> <strong>gsl_sf_gegenpoly_2</strong> <em>(double <var>lambda</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f3"></a>Function: <em>double</em> <strong>gsl_sf_gegenpoly_3</strong> <em>(double <var>lambda</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_gegenpoly_1_e</strong> <em>(double <var>lambda</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f2_005fe"></a>Function: <em>int</em> <strong>gsl_sf_gegenpoly_2_e</strong> <em>(double <var>lambda</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005f3_005fe"></a>Function: <em>int</em> <strong>gsl_sf_gegenpoly_3_e</strong> <em>(double <var>lambda</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These functions evaluate the Gegenbauer polynomials
<em>C^{(\lambda)}_n(x)</em> using explicit
representations for <em>n =1, 2, 3</em>.
</p></dd></dl>


<dl>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005fn"></a>Function: <em>double</em> <strong>gsl_sf_gegenpoly_n</strong> <em>(int <var>n</var>, double <var>lambda</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005fn_005fe"></a>Function: <em>int</em> <strong>gsl_sf_gegenpoly_n_e</strong> <em>(int <var>n</var>, double <var>lambda</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These functions evaluate the Gegenbauer polynomial <em>C^{(\lambda)}_n(x)</em> for a specific value of <var>n</var>,
<var>lambda</var>, <var>x</var> subject to <em>\lambda &gt; -1/2</em>, <em>n &gt;= 0</em>.
</p></dd></dl>


<dl>
<dt><a name="index-gsl_005fsf_005fgegenpoly_005farray"></a>Function: <em>int</em> <strong>gsl_sf_gegenpoly_array</strong> <em>(int <var>nmax</var>, double <var>lambda</var>, double <var>x</var>, double <var>result_array</var>[])</em></dt>
<dd><p>This function computes an array of Gegenbauer polynomials
<em>C^{(\lambda)}_n(x)</em> for <em>n = 0, 1, 2, \dots, nmax</em>, subject
to <em>\lambda &gt; -1/2</em>, <em>nmax &gt;= 0</em>.
</p></dd></dl>




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