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84
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 86 (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.
A Mathematica qAnalogue of Zeilberger's Algorithm for Proving qHypergeometric Identities
, 1995
"... Besides an elementary introduction to qidentities and basic hypergeometric series, a newly developed Mathematica implementation of a qanalogue of Zeilberger's fast algorithm for proving terminating qhypergeometric identities together with its theoretical background is described. To illustrate t ..."
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Cited by 62 (11 self)
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Besides an elementary introduction to qidentities and basic hypergeometric series, a newly developed Mathematica implementation of a qanalogue of Zeilberger's fast algorithm for proving terminating qhypergeometric identities together with its theoretical background is described. To illustrate the usage of the package and its range of applicability, nontrivial examples are presented as well as additional features like the computation of companion and dual identities.
Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences
, 1996
"... Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be ..."
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Cited by 50 (0 self)
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Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be performed "algorithmically". As a consequence, we can prove any identity of holonomic functions or sequences automatically. Based on this theory, the author implemented a package that contains procedures for automatic manipulations and transformations of univariate holonomic functions and sequences within the computer algebra system Mathematica. This package is introduced in detail. In addition, we describe some different techniques for proving holonomic identities.
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 (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 34 (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
Loops, matchings and alternatingsign matrices
 DISCR. MATH
, 2008
"... The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge ..."
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Cited by 31 (5 self)
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The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes on matchings is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.
HYP and HYPQ: Mathematica packages for the manipulation of binomial sums and hypergeometric series, respectively qbinomial sums and basic hypergeometric series
 Journal of Symbolic Computation
, 1995
"... Introduction Binomial series and qbinomial series, such as 1 X k=0 ` M k '` N R \Gamma k ' ; respectively 1 X k=0 q (M \Gammak)(R\Gammak) M k q N R \Gamma k q ; where the qbinomial coefficient is defined by n k q = (1 \Gamma q n )(1 \Gamma q n\Gamma1 ) \Delta ..."
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Cited by 30 (10 self)
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Introduction Binomial series and qbinomial series, such as 1 X k=0 ` M k '` N R \Gamma k ' ; respectively 1 X k=0 q (M \Gammak)(R\Gammak) M k q N R \Gamma k q ; where the qbinomial coefficient is defined by n k q = (1 \Gamma q n )(1 \Gamma q n\Gamma1 ) \Delta \Delta \Delta (1 \Gamma q n\Gammak+1 ) (1 \Gamma q k )(1 \Gamma q k\Gamma1 ) \Delta \D
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...