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<title>Gegenbauer Functions - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">7.20 Gegenbauer Functions</h3>
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<p><a name="index-Gegenbauer-functions-619"></a>
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The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
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22, where they are known as Ultraspherical polynomials. The functions
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described in this section are declared in the header file
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<samp><span class="file">gsl_sf_gegenbauer.h</span></samp>.
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— Function: double <b>gsl_sf_gegenpoly_1</b> (<var>double lambda, double x</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f1-620"></a></var><br>
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— Function: double <b>gsl_sf_gegenpoly_2</b> (<var>double lambda, double x</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f2-621"></a></var><br>
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— Function: double <b>gsl_sf_gegenpoly_3</b> (<var>double lambda, double x</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f3-622"></a></var><br>
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— Function: int <b>gsl_sf_gegenpoly_1_e</b> (<var>double lambda, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f1_005fe-623"></a></var><br>
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— Function: int <b>gsl_sf_gegenpoly_2_e</b> (<var>double lambda, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f2_005fe-624"></a></var><br>
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— Function: int <b>gsl_sf_gegenpoly_3_e</b> (<var>double lambda, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005f3_005fe-625"></a></var><br>
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<blockquote><p>These functions evaluate the Gegenbauer polynomials
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<!-- {$C^{(\lambda)}_n(x)$} -->
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C^{(\lambda)}_n(x) using explicit
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representations for n =1, 2, 3.
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<!-- Exceptional Return Values: none -->
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</p></blockquote></div>
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— Function: double <b>gsl_sf_gegenpoly_n</b> (<var>int n, double lambda, double x</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005fn-626"></a></var><br>
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— Function: int <b>gsl_sf_gegenpoly_n_e</b> (<var>int n, double lambda, double x, gsl_sf_result * result</var>)<var><a name="index-gsl_005fsf_005fgegenpoly_005fn_005fe-627"></a></var><br>
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<blockquote><p>These functions evaluate the Gegenbauer polynomial <!-- {$C^{(\lambda)}_n(x)$} -->
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C^{(\lambda)}_n(x) for a specific value of <var>n</var>,
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<var>lambda</var>, <var>x</var> subject to \lambda > -1/2, <!-- {$n \ge 0$} -->
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<!-- Domain: lambda > -1/2, n >= 0 -->
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<!-- Exceptional Return Values: GSL_EDOM -->
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</p></blockquote></div>
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— Function: int <b>gsl_sf_gegenpoly_array</b> (<var>int nmax, double lambda, double x, double result_array</var>[])<var><a name="index-gsl_005fsf_005fgegenpoly_005farray-628"></a></var><br>
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<blockquote><p>This function computes an array of Gegenbauer polynomials
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<!-- {$C^{(\lambda)}_n(x)$} -->
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C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject
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to \lambda > -1/2, <!-- {$nmax \ge 0$} -->
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<!-- Conditions: n = 0, 1, 2, ... nmax -->
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<!-- Domain: lambda > -1/2, nmax >= 0 -->
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<!-- Exceptional Return Values: GSL_EDOM -->
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</p></blockquote></div>
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