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<h1 class="chapter"> 70. zeilberger </h1>
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<h2 class="section"> 70.1 Introduction to zeilberger </h2>
<p><code>zeilberger</code> is a 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>
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<h4 class="subsubsection"> 70.1.0.1 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>
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<h4 class="subsubsection"> 70.1.0.2 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>) <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>
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<h3 class="subsection"> 70.1.1 Verbosity levels </h3>
<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:
<code>GosperVerbose</code>, <code>parGosperVeryVerbose</code>,
<code>ZeilbergerExtra</code>, <code>AntiDifferenceSummary</code>.
</p>
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<h2 class="section"> 70.2 Definitions for zeilberger </h2>
<dl>
<dt><u>Function:</u> <b>AntiDifference</b><i> (<var>F_k</var>, <var>k</var>)</i>
<a name="IDX1980"></a>
</dt>
<dd><p>Returns the hypergeometric anti-difference
of <var>F_k</var>, if it exists.
Otherwise <code>AntiDifference</code> returns <code>no_hyp_antidifference</code>.
</p></dd></dl>
<dl>
<dt><u>Function:</u> <b>Gosper</b><i> (<var>F_k</var>, <var>k</var>)</i>
<a name="IDX1981"></a>
</dt>
<dd><p>Returns the rational certificate <var>R(k)</var> for <var>F_k</var>, that is,
a rational function such that
</p>
<p><em>F_k = R(k+1) F_(k+1) - R(k) F_k</em>
</p>
<p>if it exists.
Otherwise, <code>Gosper</code> returns <code>no_hyp_sol</code>.
</p></dd></dl>
<dl>
<dt><u>Function:</u> <b>GosperSum</b><i> (<var>F_k</var>, <var>k</var>, <var>a</var>, <var>b</var>) </i>
<a name="IDX1982"></a>
</dt>
<dd><p>Returns the summmation of <var>F_k</var> from <em><var>k</var> = <var>a</var></em> to <em><var>k</var> = <var>b</var></em>
if <var>F_k</var> has a hypergeometric anti-difference.
Otherwise, <code>GosperSum</code> returns <code>nongosper_summable</code>.
</p>
<p>Examples:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) load (zeilberger);
(%o1) /usr/share/maxima/share/contrib/Zeilberger/zeilberger.mac
(%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
Dependent equations eliminated: (1)
3 n + 1
(n + -) (- 1)
2 1
(%o2) - ------------------ - -
2 4
2 (4 (n + 1) - 1)
(%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
3
- n - -
2 1
(%o3) -------------- + -
2 2
4 (n + 1) - 1
(%i4) GosperSum (x^k, k, 1, n);
n + 1
x x
(%o4) ------ - -----
x - 1 x - 1
(%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
n + 1
a! (n + 1) (- 1) a!
(%o5) - ------------------------- - ----------
a (- n + a - 1)! (n + 1)! a (a - 1)!
(%i6) GosperSum (k*k!, k, 1, n);
Dependent equations eliminated: (1)
(%o6) (n + 1)! - 1
(%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
(n + 1) (n + 2) (n + 1)!
(%o7) ------------------------ - 1
(n + 2)!
(%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
(%o8) nonGosper_summable
</pre></td></tr></table></dd></dl>
<dl>
<dt><u>Function:</u> <b>parGosper</b><i> (<var>F_{n,k}</var>, <var>k</var>, <var>n</var>, <var>d</var>)</i>
<a name="IDX1983"></a>
</dt>
<dd><p>Attempts to find a a <var>d</var>-th order recurrence for <var>F_{n,k}</var>.
</p>
<p>The algorithm yields a sequence
<em>[s_1, s_2, ..., s_m]</em> of solutions.
Each solution has the form
</p>
<p><em>[R(n, k), [a_0, a_1, ..., a_d]]</em>
</p>
<p><code>parGosper</code> returns <code>[]</code> if it fails to find a recurrence.
</p></dd></dl>
<dl>
<dt><u>Function:</u> <b>Zeilberger</b><i> (<var>F_{n,k}</var>, <var>k</var>, <var>n</var>)</i>
<a name="IDX1984"></a>
</dt>
<dd><p>Attempts to compute the indefinite hypergeometric summation of <var>F_{n,k}</var>.
</p>
<p><code>Zeilberger</code> first invokes <code>Gosper</code>, and if that fails to find a solution, then invokes
<code>parGosper</code> with order 1, 2, 3, ..., up to <code>MAX_ORD</code>.
If Zeilberger finds a solution before reaching <code>MAX_ORD</code>,
it stops and returns the solution.
</p>
<p>The algorithms yields a sequence
<em>[s_1, s_2, ..., s_m]</em> of solutions.
Each solution has the form
</p>
<p><em>[R(n,k), [a_0, a_1, ..., a_d]]</em>
</p>
<p><code>Zeilberger</code> returns <code>[]</code> if it fails to find a solution.
</p>
<p><code>Zeilberger</code> invokes <code>Gosper</code> only if <code>gosper_in_zeilberger</code> is <code>true</code>.
</p></dd></dl>
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<h2 class="section"> 70.3 General global variables </h2>
<dl>
<dt><u>Global variable:</u> <b>MAX_ORD</b>
<a name="IDX1985"></a>
</dt>
<dd><p>Default value: 5
</p>
<p><code>MAX_ORD</code> is the maximum recurrence order attempted by <code>Zeilberger</code>.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>simplified_output</b>
<a name="IDX1986"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>simplified_output</code> is <code>true</code>,
functions in the <code>zeilberger</code> package attempt
further simplification of the solution.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>linear_solver</b>
<a name="IDX1987"></a>
</dt>
<dd><p>Default value: <code>linsolve</code>
</p>
<p><code>linear_solver</code> names the solver which is used to solve the system
of equations in Zeilberger's algorithm.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>warnings</b>
<a name="IDX1988"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>warnings</code> is <code>true</code>,
functions in the <code>zeilberger</code> package print
warning messages during execution.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>gosper_in_zeilberger</b>
<a name="IDX1989"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>gosper_in_zeilberger</code> is <code>true</code>,
the <code>Zeilberger</code> function calls <code>Gosper</code> before calling <code>parGosper</code>.
Otherwise, <code>Zeilberger</code> goes immediately to <code>parGosper</code>.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>trivial_solutions</b>
<a name="IDX1990"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>trivial_solutions</code> is <code>true</code>,
<code>Zeilberger</code> returns solutions
which have certificate equal to zero, or all coefficients equal to zero.
</p></dd></dl>
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<h2 class="section"> 70.4 Variables related to the modular test </h2>
<dl>
<dt><u>Global variable:</u> <b>mod_test</b>
<a name="IDX1991"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>mod_test</code> is <code>true</code>,
<code>parGosper</code> executes a
modular test for discarding systems with no solutions.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>modular_linear_solver</b>
<a name="IDX1992"></a>
</dt>
<dd><p>Default value: <code>linsolve</code>
</p>
<p><code>modular_linear_solver</code> names the linear solver used by the modular test in <code>parGosper</code>.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>ev_point</b>
<a name="IDX1993"></a>
</dt>
<dd><p>Default value: <code>big_primes[10]</code>
</p>
<p><code>ev_point</code> is the value at which the variable <var>n</var> is evaluated
when executing the modular test in <code>parGosper</code>.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>mod_big_prime</b>
<a name="IDX1994"></a>
</dt>
<dd><p>Default value: <code>big_primes[1]</code>
</p>
<p><code>mod_big_prime</code> is the modulus used by the modular test in <code>parGosper</code>.
</p></dd></dl>
<dl>
<dt><u>Global variable:</u> <b>mod_threshold</b>
<a name="IDX1995"></a>
</dt>
<dd><p>Default value: 4
</p>
<p><code>mod_threshold</code> is the
greatest order for which the modular test in <code>parGosper</code> is attempted.
</p></dd></dl>
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