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The summation package Sigma: Underlying principles and a rhombus tiling application
 Discrete Math. Theor. Comput. Sci
, 2004
"... We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find an ..."
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Cited by 37 (28 self)
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We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find and prove a multisum identity that arose in the enumeration of rhombus tilings of a symmetric hexagon. Whereas this identity has been derived alternatively with the help of highly involved transformations of special functions, our tools enable to find and prove this identity completely automatically with the computer.
Solving parameterized linear difference equations in terms of indefinite nested sums and products
 J. Differ. Equations Appl
, 2002
"... The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. ..."
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Cited by 35 (25 self)
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The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. Keywords: symbolic summation, difference fields, ΠΣextensions, ΠΣfields AMS Subject Classification: 33FXX, 68W30, 12H10 1
A collection of denominator bounds to solve parameterized linear difference equations in ΠΣextensions
 Proc. SYNASC04, 6th Internat. Symposium on Symbolic and Numeric Algorithms for Scientific Computation
, 2004
"... An important application of solving parameterized linear dierence equations in elds, a very general class of dierence elds, is simplifying and proving of nested multisum expressions and identities. This article provides essential algorithmic building blocks that enable to search for all solutions ..."
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Cited by 32 (18 self)
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An important application of solving parameterized linear dierence equations in elds, a very general class of dierence elds, is simplifying and proving of nested multisum expressions and identities. This article provides essential algorithmic building blocks that enable to search for all solutions of such dierence equations. More precisely, these algorithms allow to exploit a denominator elimination strategy which amounts to look for solutions in a polynomial ring instead of searching for rational function solutions. 1.
Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣfields
 Appl. Algebra Engrg. Comm. Comput
"... An important application of solving parameterized linear difference equations in ΠΣfields, a very general class of difference fields, is simplifying and proving of nested multisum expressions and identities. Together with other reduction techniques described elsewhere, the algorithms considered in ..."
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Cited by 22 (15 self)
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An important application of solving parameterized linear difference equations in ΠΣfields, a very general class of difference fields, is simplifying and proving of nested multisum expressions and identities. Together with other reduction techniques described elsewhere, the algorithms considered in this article can be used to search for all solutions of such difference equations. More precisely, within a typical reduction step one often is faced with subproblems to find all solutions of linear difference equations where the solutions live in a polynomial ring. The algorithms under consideration deliver degree bounds for these polynomial solutions. 1.
A new Sigma approach to multisummation
 Adv. Appl. Math
, 2005
"... Abstract. We present a general algorithmic framework that allows not only to deal with summation problems over summands being rational expressions in indenite nested sums and products (Karr 1981), but also over @nite and holonomic summand expressions that are given by a linear recurrence. This appr ..."
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Cited by 17 (13 self)
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Abstract. We present a general algorithmic framework that allows not only to deal with summation problems over summands being rational expressions in indenite nested sums and products (Karr 1981), but also over @nite and holonomic summand expressions that are given by a linear recurrence. This approach implies new computer algebra tools implemented in Sigma to solve multisummation problems eciently. For instance, the extended Sigma package has been applied successively to provide a computerassisted proof of Stembridge's TSPP theorem. 1.
Polynomial and rational solutions of holonomic systems
 n o 12
"... Polynomial and rational solutions for linear ordinary differential equations can be obtained by algorithmic methods. For instance, the maple package DEtools provides efficient functions polysols and ratsols to find polynomial and rational ..."
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Cited by 16 (4 self)
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Polynomial and rational solutions for linear ordinary differential equations can be obtained by algorithmic methods. For instance, the maple package DEtools provides efficient functions polysols and ratsols to find polynomial and rational
SYMBOLIC SUMMATION ASSISTS COMBINATORICS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 56 (2007), ARTICLE B56B
, 2007
"... We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems. ..."
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Cited by 15 (8 self)
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We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems.
A refined difference field theory for symbolic summation
 Journal of Symbolic Computation
, 2008
"... In this article we present a refined summation theory based on Karr’s difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to evaluate Feynman integrals from Perturbative Quantum Field Theo ..."
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Cited by 13 (8 self)
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In this article we present a refined summation theory based on Karr’s difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to evaluate Feynman integrals from Perturbative Quantum Field Theory. Key words: Symbolic summation, difference fields, nested depth
Fast algorithms for polynomial solutions of linear differential equations
 In Proceedings of ISSAC’05
, 2005
"... Si l’on se bornait à demander les intégrales entières, le problème n’offrirait aucune difficulté. 1 Joseph Liouville, 1833. We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are ..."
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Cited by 13 (4 self)
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Si l’on se bornait à demander les intégrales entières, le problème n’offrirait aucune difficulté. 1 Joseph Liouville, 1833. We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in an integer N (that can be computed from the equation), even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(N log 3 N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O ( √ N log 2 N) bit operations. In general, the integer N is not bounded polynomially in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.
A fast approach to creative telescoping
 Mathematics in Computer Science
, 2010
"... Abstract. In this note we reinvestigate the task of computing creative telescoping relations in differentialdifference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of ..."
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Cited by 11 (7 self)
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Abstract. In this note we reinvestigate the task of computing creative telescoping relations in differentialdifference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.