(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7314, 258]*) (*NotebookOutlinePosition[ 7945, 280]*) (* CellTagsIndexPosition[ 7901, 276]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["The Simplified Zeilberger Algorithm", Background->RGBColor[0, 0, 1]]], "Title"], Cell[TextData[{ "implemented in a ", StyleBox["Mathematica ", FontSlant->"Italic"], "package by \nAndrew Sills, Rutgers University\n", StyleBox["version of January 24, 2006.", FontSize->14] }], "Subtitle"], Cell[BoxData[ StyleBox["Introduction", "Section"]], "Input"], Cell[TextData[{ "Zeilberger's algorithm (also known as the creative telescoping algorithm) \ finds a recurrence of the form\n", StyleBox["\[Sum]", FontSize->24], Cell[BoxData[ \(TraditionalForm\`\_\(i = 0\)\%L\)], FontSize->24], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["e", FontSize->24], "i"], TraditionalForm]]], StyleBox["(", FontSize->24], StyleBox["n", FontSize->24, FontSlant->"Italic"], StyleBox[") ", FontSize->24], StyleBox["F", FontSize->24, FontSlant->"Italic"], StyleBox["(", FontSize->24], StyleBox["n", FontSize->24, FontSlant->"Italic"], StyleBox["+", FontSize->24], StyleBox["i", FontSize->24, FontSlant->"Italic"], StyleBox[",", FontSize->24], StyleBox["k", FontSize->24, FontSlant->"Italic"], StyleBox[") = ", FontSize->24], StyleBox["G", FontSize->24, FontSlant->"Italic"], StyleBox["(", FontSize->24], StyleBox["n,k+", FontSize->24, FontSlant->"Italic"], StyleBox["1) - ", FontSize->24], StyleBox["G", FontSize->24, FontSlant->"Italic"], StyleBox["(", FontSize->24], StyleBox["n,k", FontSize->24, FontSlant->"Italic"], StyleBox[") (*)\n", FontSize->24], "where ", StyleBox["F", FontSlant->"Italic"], " is a hypergeometric term in ", StyleBox["n", FontSlant->"Italic"], " and ", StyleBox["k ", FontSlant->"Italic"], "with finite support and ", StyleBox["G ", FontSlant->"Italic"], "is a rational function multiple of ", StyleBox["F", FontSlant->"Italic"], ".\n\nThe original Zeilberger algorithm is discussed in\n\t[1] D. \ Zeilberger, A fast algorithm for proving terminating hypergeometric \ identities, ", StyleBox["Discrete Math.", FontSlant->"Italic"], " ", StyleBox["80", FontWeight->"Bold"], " (1990), 207-211.\n\t[2] D. Zeilberger, The method of creative \ telescoping, ", StyleBox["J. Symbolic Comput", FontSlant->"Italic"], ". ", StyleBox["11", FontWeight->"Bold"], " (1991), 195-204.\n\t[3] M. Petkovsek, H.S. Wilf and D. Zeilberger, ", StyleBox["A=B, ", FontSlant->"Italic"], "A.K. Peters, 1996.\n\nA simplified version of the algorithm (implemented \ in the ", StyleBox["Mathematica", FontSlant->"Italic"], " package NewZeil.m) is implied by the theoretical results in\n\t[4] M. \ Mohammed and D. Zeilberger, Sharp upper bounds for the orders of the \ recurrences output by the Zeilberger and ", StyleBox["q", FontSlant->"Italic"], "-Zeilberger algorithms, ", StyleBox["J. Symbolic Comput. ", FontSlant->"Italic"], StyleBox["39", FontWeight->"Bold"], "(2005), 201-207.", "\n" }], "Text"], Cell[CellGroupData[{ Cell["Examples", "Section"], Cell["First, load the package:", "Text"], Cell[BoxData[ \(<< NewZeil.m\)], "Input"], Cell[BoxData[""], "Input"], Cell["\<\ Consider the following examples. The output is of the form { \ Recurrence, Certificate }, where \"Recurrence\" is recurrence found by the \ algorithm from the left hand side of (*) above and \"Certificate\" is the \ rational certificate function (which equals G/F from (*) above).\ \>", "Text"], Cell[BoxData[""], "Input"], Cell[TextData[{ "The input format of the main function, Zeil, is\nZeil[ H, z, POL, sum_var, \ discrete_var, {params}], where\n\tH is the 'pure hypergeometric part', i.e. \ ratios of Pochhammer symbols, binomial coefficients, and factorials;\n\tz is \ the hypergeometric argument;\n\tsum_var is a symbol to represent the \ summation index (usually denoted by ", StyleBox["k", FontSlant->"Italic"], " in the WZ literature);\n\tdiscrete_var is a symbol to represent the \ discrete parameter (usually denoted by ", StyleBox["n", FontSlant->"Italic"], " in the WZ literature);\n\tshift_op is is a symbol to represent the shift \ operator (usually deonted by ", StyleBox["N", FontSlant->"Italic"], " in the WZ literature); and\n\t{params} is an optional list of additional \ parameters.\nConsider the following examples. " }], "Text"], Cell[BoxData[ \(Zeil[\ Binomial[n, k], \ 1, 1, k, n]\)], "Input"], Cell[BoxData[ \(Zeil[\ Binomial[n, k], z, 1, k, n]\)], "Input"], Cell[BoxData[ \(Zeil[\ Binomial[n, k]^2, \ 1, 1, k, n]\)], "Input"], Cell[BoxData[ \(Zeil[\ Binomial[n, k]^3, 1, 1, k, n]\)], "Input"], Cell[BoxData[ \(Zeil[\ Pochhammer[a, k]\ Pochhammer[n, k], \ 1, 1, k, n, {a}]\)], "Input"], Cell[BoxData[ \(Zeil[\ 1/\(k!\)^3\ \ Pochhammer[n, k], 1, n^3 + k, k, n]\)], "Input"], Cell[BoxData[ \(Zeil[Binomial[x, k]\ Binomial[a, n - k], 1, 1, k, n, {x, a}]\)], "Input"], Cell["\<\ If the global variable 'Verbose' is set to 'True', the results of \ intermediate calculatations are displayed using the notation of [4], allowing \ the user to follow the details of the simplified Zeilberger algorithm.\ \>", \ "Text"], Cell[BoxData[ \(Verbose = True\)], "Input"], Cell[BoxData[ \(Zeil[\ Pochhammer[a, k]\ Pochhammer[n, k], \ 1, 1, k, n, {a}]\)], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for X", ScreenRectangle->{{0, 1680}, {0, 1050}}, WindowSize->{754, 607}, WindowMargins->{{Automatic, 147}, {Automatic, 80}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 105, 1, 92, "Title"], Cell[1884, 56, 221, 7, 98, "Subtitle"], Cell[2108, 65, 70, 2, 38, "Input"], Cell[2181, 69, 2758, 109, 278, "Text"], Cell[CellGroupData[{ Cell[4964, 182, 27, 0, 73, "Section"], Cell[4994, 184, 40, 0, 32, "Text"], Cell[5037, 186, 45, 1, 27, "Input"], Cell[5085, 189, 26, 0, 27, "Input"], Cell[5114, 191, 305, 5, 68, "Text"], Cell[5422, 198, 26, 0, 27, "Input"], Cell[5451, 200, 854, 18, 176, "Text"], Cell[6308, 220, 69, 1, 27, "Input"], Cell[6380, 223, 67, 1, 27, "Input"], Cell[6450, 226, 71, 1, 27, "Input"], Cell[6524, 229, 69, 1, 27, "Input"], Cell[6596, 232, 101, 2, 27, "Input"], Cell[6700, 236, 89, 1, 27, "Input"], Cell[6792, 239, 93, 1, 27, "Input"], Cell[6888, 242, 244, 5, 50, "Text"], Cell[7135, 249, 47, 1, 27, "Input"], Cell[7185, 252, 101, 2, 27, "Input"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)