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211
Analytic Analysis of Algorithms
, 1992
"... . The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial struct ..."
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Cited by 331 (13 self)
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. The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial structures. Asymptotic methods based on complex analysis permit to extract directly coefficients of structurally complicated generating functions without a need for explicit coefficient expansions. Three major groups of problems relative to algebraic equations, differential equations, and iteration are presented. The range of applications includes formal languages, tree enumerations, comparisonbased searching and sorting, digital structures, hashing and occupancy problems. These analytic approaches allow an abstract discussion of asymptotic properties of combinatorial structures and schemas while opening the way for automatic analysis of whole classes of combinatorial algorithms. I...
GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable
, 1992
"... We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package. ..."
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Cited by 165 (18 self)
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We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.
A Mathematica Version of Zeilberger's Algorithm for Proving Binomial Coefficient Identities
, 1993
"... ..."
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 100 (12 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
An Extension Of Zeilberger's Fast Algorithm To General Holonomic Functions
 DISCRETE MATH
, 2000
"... We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy. ..."
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Cited by 90 (5 self)
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We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.
Hypergeometric solutions of linear recurrences with polynomial coefficients
 J. Symb. Comput
, 1992
"... We describe algorithm Hyper which can be used to find all hypergeometric solutions of linear recurrences with polynomial coefficients. Let K be a field of characteristic zero. We assume that K is computable, meaning that the elements of K can be finitely represented and that there exist algorithms f ..."
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Cited by 79 (10 self)
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We describe algorithm Hyper which can be used to find all hypergeometric solutions of linear recurrences with polynomial coefficients. Let K be a field of characteristic zero. We assume that K is computable, meaning that the elements of K can be finitely represented and that there exist algorithms for carrying out the field operations. Let KN denote the ring of all sequences over K, with addition and multiplication
Rational functions certify combinatorial identities
 J. Amer. Math. Soc
, 1990
"... This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present acerti cate that consists of a pair of functions of two integer variables. To prove the identity, take the two functions that are given, check that condition (1) below is s ..."
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Cited by 78 (6 self)
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(Show Context)
This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present acerti cate that consists of a pair of functions of two integer variables. To prove the identity, take the two functions that are given, check that condition (1) below is satis ed (a simple mechanical task), and check the equally simple fact that the boundary conditions (F1), (G1), (G2) below
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 55 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.