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.

Abstract. An important application of solving parameterized linear difference equations in ΠΣ-fields, a very general class of difference fields, is simplifying of multi-sum expressions and proving of multi-sum identities. This article provides essential algorithmic building blocks that allow one to search for all solutions of such difference equations. More precisely, these algorithms enable one to exploit a denominator elimination strategy which amounts to look for solutions in a polynomial ring instead of searching for rational function solutions.

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

An important application of solving parameterized linear difference equations in ΠΣ-fields, 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.

We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multi-sum examples which are related to combinatorial problems.

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 multi-summation problems efficiently. For instance, the extended Sigma package has been applied successively to provide a computer-assisted proof of Stembridge’s TSPP theorem. 1.

This manual describes the functionality of the Mathematica package HolonomicFunctions. It is a very powerful tool for the work with special functions, it can assist in solving summation and integration problems, it can automatically prove special function identities, and much more. The package has been developed in the frame of the PhD thesis [8]. The whole theory and the algorithms are described there, and it contains also many references for further reading as well as some more advanced examples; the examples in this manual are mostly of a very simple nature in order to illustrate clearly the use of the software. HolonomicFunctions is freely available from the RISC combinatorics software webpage www.risc.uni-linz.ac.at/research/combinat/software/HolonomicFunctions/ Short references Annihilator [ expr, ops] computes annihilating operators for the expression expr with respect to the Ore operators ops (p. 5). AnnihilatorDimension [ ann] gives the dimension of the annihilating left ideal

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

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.

We provide a new algorithm for indefinite nested summation which is applicable to summands involving unspecified sequences x(n). More than that, we show how to extend Karr’s algorithm to a general summation framework by which additional types of summand expressions can be handled. Our treatment of unspecified sequences can be seen as a first illustrative application of this approach.