1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>Maxima Manual: Functions and Variables for zeilberger</title>
<meta name="description" content="Maxima Manual: Functions and Variables for zeilberger">
<meta name="keywords" content="Maxima Manual: Functions and Variables for zeilberger">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<link href="maxima_toc.html#Top" rel="start" title="Top">
<link 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" rel="index" title="Указатель функций и переменных">
<link href="maxima_toc.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="maxima_261.html#zeilberger_002dpkg" rel="up" title="zeilberger-pkg">
<link 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" rel="next" title="Указатель функций и переменных">
<link href="maxima_262.html#Introduction-to-zeilberger" rel="previous" title="Introduction to zeilberger">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
body {color: black; background: white; margin-left: 8%; margin-right: 13%;
font-family: "FreeSans", sans-serif}
h1 {font-size: 150%; font-family: "FreeSans", sans-serif}
h2 {font-size: 125%; font-family: "FreeSans", sans-serif}
h3 {font-size: 100%; font-family: "FreeSans", sans-serif}
a[href] {color: rgb(0,0,255); text-decoration: none;}
a[href]:hover {background: rgb(220,220,220);}
div.textbox {border: solid; border-width: thin; padding-top: 1em;
padding-bottom: 1em; padding-left: 2em; padding-right: 2em}
div.titlebox {border: none; padding-top: 1em; padding-bottom: 1em;
padding-left: 2em; padding-right: 2em; background: rgb(200,255,255);
font-family: sans-serif}
div.synopsisbox {
border: none; padding-top: 1em; padding-bottom: 1em; padding-left: 2em;
padding-right: 2em; background: rgb(255,220,255);}
pre.example {border: 1px solid rgb(180,180,180); padding-top: 1em;
padding-bottom: 1em; padding-left: 1em; padding-right: 1em;
background-color: rgb(238,238,255)}
div.spacerbox {border: none; padding-top: 2em; padding-bottom: 2em}
div.image {margin: 0; padding: 1em; text-align: center}
div.categorybox {border: 1px solid gray; padding-top: 1em; padding-bottom: 1em;
padding-left: 1em; padding-right: 1em; background: rgb(247,242,220)}
img {max-width:80%; max-height: 80%; display: block; margin-left: auto; margin-right: auto}
-->
</style>
<link rel="icon" href="figures/favicon.ico">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6>"></script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
</head>
<body lang="ru" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="Functions-and-Variables-for-zeilberger"></a>
<div class="header">
<p>
Previous: <a href="maxima_262.html#Introduction-to-zeilberger" accesskey="p" rel="previous">Introduction to zeilberger</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="Functions-and-Variables-for-zeilberger-1"></a>
<h3 class="section">73.2 Functions and Variables for zeilberger</h3>
<a name="AntiDifference"></a><a name="Item_003a-zeilberger_002fdeffn_002fAntiDifference"></a><dl>
<dt><a name="index-AntiDifference"></a>Function: <strong>AntiDifference</strong> <em>(<em>F_k</em>, <var>k</var>)</em></dt>
<dd>
<p>Returns the hypergeometric anti-difference of <em>F_k</em>, if it exists.<br>
Otherwise <code>AntiDifference</code> returns <code>no_hyp_antidifference</code>.
</p>
</dd></dl>
<a name="Gosper"></a><a name="Item_003a-zeilberger_002fdeffn_002fGosper"></a><dl>
<dt><a name="index-Gosper"></a>Function: <strong>Gosper</strong> <em>(<em>F_k</em>, <var>k</var>)</em></dt>
<dd><p>Returns the rational certificate <em>R(k)</em> for <em>F_k</em>, that is,
a rational function such
that
\(F_k = R\left(k+1\right) \, F_{k+1} - R\left(k\right) \, F_k,\) if it exists.
Otherwise, <code>Gosper</code> returns <code>no_hyp_sol</code>.
</p>
</dd></dl>
<a name="GosperSum"></a><a name="Item_003a-zeilberger_002fdeffn_002fGosperSum"></a><dl>
<dt><a name="index-GosperSum"></a>Function: <strong>GosperSum</strong> <em>(<em>F_k</em>, <var>k</var>, <var>a</var>, <var>b</var>)</em></dt>
<dd>
<p>Returns the summation of <em>F_k</em> from <em><var>k</var> = <var>a</var></em> to <em><var>k</var> = <var>b</var></em>
if <em>F_k</em> has a hypergeometric anti-difference.
Otherwise, <code>GosperSum</code> returns <code>nongosper_summable</code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) load ("zeilberger")$
</pre><pre class="example">(%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)
</pre><pre class="example">(%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
3
- n - -
2 1
(%o3) -------------- + -
2 2
4 (n + 1) - 1
</pre><pre class="example">(%i4) GosperSum (x^k, k, 1, n);
n + 1
x x
(%o4) ------ - -----
x - 1 x - 1
</pre><pre class="example">(%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)!
</pre><pre class="example">(%i6) GosperSum (k*k!, k, 1, n);
Dependent equations eliminated: (1)
(%o6) (n + 1)! - 1
</pre><pre class="example">(%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
(n + 1) (n + 2) (n + 1)!
(%o7) ------------------------ - 1
(n + 2)!
</pre><pre class="example">(%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
(%o8) NON_GOSPER_SUMMABLE
</pre></div>
</dd></dl>
<a name="parGosper"></a><a name="Item_003a-zeilberger_002fdeffn_002fparGosper"></a><dl>
<dt><a name="index-parGosper"></a>Function: <strong>parGosper</strong> <em>(<em>F_(n,k)</em>, <var>k</var>, <var>n</var>, <var>d</var>)</em></dt>
<dd>
<p>Attempts to find a <var>d</var>-th order recurrence for <em>F_(n,k)</em>.
</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>
<a name="Zeilberger"></a><a name="Item_003a-zeilberger_002fdeffn_002fZeilberger"></a><dl>
<dt><a name="index-Zeilberger"></a>Function: <strong>Zeilberger</strong> <em>(<em>F_(n,k)</em>, <var>k</var>, <var>n</var>)</em></dt>
<dd>
<p>Attempts to compute the indefinite hypergeometric summation of <em>F_(n,k)</em>.
</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>
<a name="General-global-variables"></a>
<h3 class="section">73.3 General global variables</h3>
<a name="MAX_005fORD"></a><a name="Item_003a-zeilberger_002fdefvr_002fMAX_005fORD"></a><dl>
<dt><a name="index-MAX_005fORD"></a>Global variable: <strong>MAX_ORD</strong></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>
<a name="simplified_005foutput"></a><a name="Item_003a-zeilberger_002fdefvr_002fsimplified_005foutput"></a><dl>
<dt><a name="index-simplified_005foutput"></a>Global variable: <strong>simplified_output</strong></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>
<a name="linear_005fsolver"></a><a name="Item_003a-zeilberger_002fdefvr_002flinear_005fsolver"></a><dl>
<dt><a name="index-linear_005fsolver"></a>Global variable: <strong>linear_solver</strong></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>
<a name="warnings"></a><a name="Item_003a-zeilberger_002fdefvr_002fwarnings"></a><dl>
<dt><a name="index-warnings"></a>Global variable: <strong>warnings</strong></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>
<a name="Gosper_005fin_005fZeilberger"></a><a name="Item_003a-zeilberger_002fdefvr_002fGosper_005fin_005fZeilberger"></a><dl>
<dt><a name="index-Gosper_005fin_005fZeilberger"></a>Global variable: <strong>Gosper_in_Zeilberger</strong></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>
<a name="trivial_005fsolutions"></a><a name="Item_003a-zeilberger_002fdefvr_002ftrivial_005fsolutions"></a><dl>
<dt><a name="index-trivial_005fsolutions"></a>Global variable: <strong>trivial_solutions</strong></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>
<a name="Variables-related-to-the-modular-test"></a>
<h3 class="section">73.4 Variables related to the modular test</h3>
<a name="mod_005ftest"></a><a name="Item_003a-zeilberger_002fdefvr_002fmod_005ftest"></a><dl>
<dt><a name="index-mod_005ftest"></a>Global variable: <strong>mod_test</strong></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>
<a name="modular_005flinear_005fsolver"></a><a name="Item_003a-zeilberger_002fdefvr_002fmodular_005flinear_005fsolver"></a><dl>
<dt><a name="index-modular_005flinear_005fsolver"></a>Global variable: <strong>modular_linear_solver</strong></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>
<a name="ev_005fpoint"></a><a name="Item_003a-zeilberger_002fdefvr_002fev_005fpoint"></a><dl>
<dt><a name="index-ev_005fpoint"></a>Global variable: <strong>ev_point</strong></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>
<a name="mod_005fbig_005fprime"></a><a name="Item_003a-zeilberger_002fdefvr_002fmod_005fbig_005fprime"></a><dl>
<dt><a name="index-mod_005fbig_005fprime"></a>Global variable: <strong>mod_big_prime</strong></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>
<a name="mod_005fthreshold"></a><a name="Item_003a-zeilberger_002fdefvr_002fmod_005fthreshold"></a><dl>
<dt><a name="index-mod_005fthreshold"></a>Global variable: <strong>mod_threshold</strong></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>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_262.html#Introduction-to-zeilberger" accesskey="p" rel="previous">Introduction to zeilberger</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>
</body>
</html>
|
