... 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.

Besides an elementary introduction to q-identities and basic hypergeometric series, a newly developed Mathematica implementation of a q-analogue of Zeilberger's fast algorithm for proving terminating q-hypergeometric identities together with its theoretical background is described. To illustrate the usage of the package and its range of applicability, non-trivial examples are presented as well as additional features like the computation of companion and dual identities.

Abstract not found

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.

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.

The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣ-fields, 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, ΠΣ-extensions, ΠΣ-fields AMS Subject Classification: 33FXX, 68W30, 12H10 1

Introduction Binomial series and q-binomial 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 q-binomial 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

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. 1

Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.

New short and easy computer proofs of finite versions of the Rogers-Ramanujan identities and of similar type are given. These include a very short proof of the first Rogers-Ramanujan identity that was missed by computers, and a new proof of the well-known quintuple product identity by creative telescoping. AMS Subject Classification. 05A19; secondary 11B65, 05A17 1 Introduction The celebrated Rogers-Ramanujan identities stated as series-product 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 well-known that number theoretic identities like these, or of similar type, can be deduced as limiting cases of q-hypergeometric finite-sum identities. Due to recent algorithmic breakthroughs, see for instance Zeilberger [24], or, Wilf and Zeilberger [23], proving th...