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<a name="Introduction-to-zeilberger"></a>
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<p>
Next: <a href="maxima_332.html#Functions-and-Variables-for-zeilberger" accesskey="n" rel="next">Functions and Variables for zeilberger</a>, Previous: <a href="maxima_330.html#zeilberger" accesskey="p" rel="previous">zeilberger</a>, Nach oben: <a href="maxima_330.html#zeilberger" accesskey="u" rel="up">zeilberger</a> [<a href="maxima_toc.html#SEC_Contents" title="Inhaltsverzeichnis" rel="contents">Inhalt</a>][<a href="maxima_334.html#Index-der-Variablen-und-Funktionen" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-zeilberger-1"></a>
<h3 class="section">77.1 Introduction to zeilberger</h3>
<p><code>zeilberger</code> is an implementation of Zeilberger’s algorithm for definite
hypergeometric summation, and also Gosper’s algorithm for indefinite
hypergeometric summation. <code>zeilberger</code> makes use of the "filtering"
optimization method developed by Axel Riese. <code>zeilberger</code> was developed
by Fabrizio Caruso. <code>load("zeilberger")</code> loads this package.
</p>
<a name="The-indefinite-summation-problem"></a>
<h4 class="subsection">77.1.1 The indefinite summation problem</h4>
<p><code>zeilberger</code> implements Gosper’s algorithm for indefinite hypergeometric
summation. Given a hypergeometric term <em>F_k</em> in <em>k</em> we want to find
its hypergeometric anti-difference, that is, a hypergeometric term <em>f_k</em>
such that
<em>F_k = f_(k+1) - f_k</em>.
</p>
<a name="The-definite-summation-problem"></a>
<h4 class="subsection">77.1.2 The definite summation problem</h4>
<p><code>zeilberger</code> implements Zeilberger’s algorithm for definite hypergeometric
summation. Given a proper hypergeometric term (in <em>n</em> and <em>k</em>)
</p>
<p><em>F_(n,k)</em>
and a positive integer <em>d</em> we want to find a <em>d</em>-th order linear
recurrence with polynomial coefficients (in <em>n</em>) for
<em>F_(n,k)</em>
and a rational function <em>R</em> in <em>n</em> and <em>k</em> such that
</p>
<p><em>a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k))</em>,
</p>
<p>where
<em>Delta_k</em>
is the <em>k</em>-forward difference operator, i.e.,
<em>Delta_k(t_k) := t_(k+1) - t_k</em>.
</p>
<a name="Verbosity-levels"></a>
<h4 class="subsection">77.1.3 Verbosity levels</h4>
<p>There are also verbose versions of the commands which are called by adding one
of the following prefixes:
</p>
<dl compact="compact">
<dt><code>Summary</code></dt>
<dd><p>Just a summary at the end is shown
</p></dd>
<dt><code>Verbose</code></dt>
<dd><p>Some information in the intermidiate steps
</p></dd>
<dt><code>VeryVerbose</code></dt>
<dd><p>More information
</p></dd>
<dt><code>Extra</code></dt>
<dd><p>Even more information including information on the linear system in
Zeilberger’s algorithm
</p></dd>
</dl>
<p>For example:
</p><p align="left"><code>GosperVerbose</code>, <code>parGosperVeryVerbose</code>, <code>ZeilbergerExtra</code>,
<code>AntiDifferenceSummary</code>.
</p>
<a name="Item_003a-zeilberger_002ede_002fnode_002fFunctions-and-Variables-for-zeilberger"></a><hr>
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Next: <a href="maxima_332.html#Functions-and-Variables-for-zeilberger" accesskey="n" rel="next">Functions and Variables for zeilberger</a>, Previous: <a href="maxima_330.html#zeilberger" accesskey="p" rel="previous">zeilberger</a>, Nach oben: <a href="maxima_330.html#zeilberger" accesskey="u" rel="up">zeilberger</a> [<a href="maxima_toc.html#SEC_Contents" title="Inhaltsverzeichnis" rel="contents">Inhalt</a>][<a href="maxima_334.html#Index-der-Variablen-und-Funktionen" title="Index" rel="index">Index</a>]</p>
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