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41
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 90 (9 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 65 (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.
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 39 (27 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 37 (24 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
qHypergeometric Solutions of qDifference Equations
"... We present algorithm qHyper for finding all solutions y(x) of a linear homogeneous qdifference equation, such that y(qx) = r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and definite qhypergeometric summation in closed ..."
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Cited by 27 (3 self)
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We present algorithm qHyper for finding all solutions y(x) of a linear homogeneous qdifference equation, such that y(qx) = r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and definite qhypergeometric summation in closed form. The research described in this publication was made possible in part by Grant J12100 from the International Science Foundation and Russian Government. y Supported in part by grant P7720 of the Austrian FWF. z Supported in part by grant J26193010194 of the Slovenian Ministry of Science and Technology. 1 1 Introduction As a motivating example, consider the following secondorder qdifference equation y n+2 \Gamma (1 + q) x y n+1 + x 2 y n = 0 (1) where x = q n . This is a homogeneous linear equation with coefficients which are polynomials in x. It is easy to check that y (1) n = q ( n 2 ) and y (2) n = q ( n 2 ) \Gamman both solve (1). Note that their consec...
Short and Easy Computer Proofs of the RogersRamanujan Identities and of Identities of Similar Type
, 1994
"... New short and easy computer proofs of finite versions of the RogersRamanujan identities and of similar type are given. These include a very short proof of the first RogersRamanujan identity that was missed by computers, and a new proof of the wellknown quintuple product identity by creative teles ..."
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Cited by 24 (4 self)
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New short and easy computer proofs of finite versions of the RogersRamanujan identities and of similar type are given. These include a very short proof of the first RogersRamanujan identity that was missed by computers, and a new proof of the wellknown quintuple product identity by creative telescoping. AMS Subject Classification. 05A19; secondary 11B65, 05A17 1 Introduction The celebrated RogersRamanujan identities stated as seriesproduct identities are 1 + 1 X k=1 q k 2 +ak (1 \Gamma q)(1 \Gamma q 2 ) \Delta \Delta \Delta (1 \Gamma q k ) = 1 Y j=0 1 (1 \Gamma q 5j+a+1 )(1 \Gamma q 5j \Gammaa+4 ) (1) where a = 0 or a = 1, see e.g. Andrews [6] which also contains a brief historical account. It is wellknown that number theoretic identities like these, or of similar type, can be deduced as limiting cases of qhypergeometric finitesum identities. Due to recent algorithmic breakthroughs, see for instance Zeilberger [24], or, Wilf and Zeilberger [23], proving th...
qMultiSum  A Package for Proving qHypergeometric Multiple Summation Identities
 JOURNAL OF SYMBOLIC COMPUTATION
"... A Mathematica package for finding recurrences for qhypergeometric multiple sums is introduced. Together with a detailed description of the theoretical background, we present several examples to illustrate its usage and range of applicability. In particular, various computer proofs of recently disco ..."
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Cited by 20 (3 self)
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A Mathematica package for finding recurrences for qhypergeometric multiple sums is introduced. Together with a detailed description of the theoretical background, we present several examples to illustrate its usage and range of applicability. In particular, various computer proofs of recently discovered identities are exhibited.
The colored Jones function is qholonomic
, 2005
"... A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, ..."
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Cited by 19 (5 self)
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A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3–space, we prove from first principles that the colored Jones function is a multisum of a q–properhypergeometric function, and thus it is q–holonomic. We demonstrate our results by computer calculations.
A new Sigma approach to multisummation
 the Dave Robbins memorial issue of Advances in Applied Math
, 2005
"... Abstract. We present a general algorithmic framework that allows not only to deal with summation problems over summands being rational expressions in indefinite nested sums and products (Karr 1981), but also over ∂finite and holonomic summand expressions that are given by a linear recurrence. This ..."
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Cited by 17 (12 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 indefinite nested sums and products (Karr 1981), but also over ∂finite 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 efficiently. For instance, the extended Sigma package has been applied successively to provide a computerassisted proof of Stembridge’s TSPP theorem. 1.
Symbolic Summation  Some Recent Developments
 Computer Algebra in Science and Engineering  Algorithms, Systems, and Applications
, 1995
"... In recent years, the problem of symbolic summation has received much attention due to the exciting applications of Zeilberger's method for definite hypergeometric summation. This lead to renewed interest in the central part of the algorithmic machinery, Gosper's "classical" method for indefinite hyp ..."
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Cited by 16 (2 self)
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In recent years, the problem of symbolic summation has received much attention due to the exciting applications of Zeilberger's method for definite hypergeometric summation. This lead to renewed interest in the central part of the algorithmic machinery, Gosper's "classical" method for indefinite hypergeometric summation. We review some of the recent (partly unpublished as yet) work done in this field, with a particular emphasis on the unifying and guiding role of normal forms for polynomials and rational functions, especially adapted to summation algorithms. 1. Introduction The algorithmic problem of symbolic summation can be most easily introduced by presenting it as the discrete analogue of the wellknown problem of "symbolic integration" or "integration in finite terms". Without being too formal, we can simply say that the role of the differential operator in the latter problem is taken over by the difference operator \Delta: (\Delta g) (x) := g(x + 1) \Gamma g(x) ; where g is a f...