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<a name="Introduction-to-zeilberger"></a>
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<p>
Next: <a href="maxima_263.html#Functions-and-Variables-for-zeilberger" accesskey="n" rel="next">Functions and Variables for zeilberger</a>, Previous: <a href="maxima_261.html#zeilberger_002dpkg" accesskey="p" rel="previous">zeilberger-pkg</a>, Up: <a href="maxima_261.html#zeilberger_002dpkg" accesskey="u" rel="up">zeilberger-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-zeilberger-1"></a>
<h3 class="section">73.1 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.
</p>
<p><code>zeilberger</code> makes use of the "filtering" optimization method developed by Axel Riese.
</p>
<p><code>zeilberger</code> was developed by Fabrizio Caruso.
</p>
<p><code>load ("zeilberger")</code> loads this package.
</p>
<a name="The-indefinite-summation-problem"></a>
<h4 class="subsection">73.1.1 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
</p>
$$
F_k = f_{k+1} - f_k.
$$
<a name="The-definite-summation-problem"></a>
<h4 class="subsection">73.1.2 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>)
\(F_{n,k}\) 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
\(F_{n,k}\) and
a rational function <em>R</em> in <em>n</em> and <em>k</em> such that
</p>
$$
a_0 \, F_{n,k} + \ldots + a_d \, F_{n+d,k} = \Delta_K \left(R\left(n,k\right) F_{n,k}\right),
$$
<p>where
\(\Delta_k\) is the <em>k</em>-forward difference
operator, i.e.,
\(\Delta_k \left(t_k\right) \equiv t_{k+1} - t_k.\)</p>
<a name="Verbosity-levels"></a>
<h4 class="subsection">73.1.3 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 intermediate 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:<br>
<code>GosperVerbose</code>, <code>parGosperVeryVerbose</code>,
<code>ZeilbergerExtra</code>, <code>AntiDifferenceSummary</code>.
</p>
<a name="Item_003a-zeilberger_002fnode_002fFunctions-and-Variables-for-zeilberger"></a><hr>
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Next: <a href="maxima_263.html#Functions-and-Variables-for-zeilberger" accesskey="n" rel="next">Functions and Variables for zeilberger</a>, Previous: <a href="maxima_261.html#zeilberger_002dpkg" accesskey="p" rel="previous">zeilberger-pkg</a>, Up: <a href="maxima_261.html#zeilberger_002dpkg" accesskey="u" rel="up">zeilberger-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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