Abstract not found

Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a non-zero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic extension of a transcendental liouvillian extension of the field of rational functions. We show how one can use this result to give a procedure to find a basis for the space of solutions, liouvillian over K, of L(y)=0 where K is such a field and L(y) has coefficients in K. 1.

”I could never resist a definite integral ” (G. H. Hardy [Co]) ”One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high-school physics teacher Mr. Bader had given me.... The book also showed how to differentiate parameters under the integral sign-It’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me. ” (Richard P. Feynman[F])

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

Abstract. We use C. Schneider’s summation software Sigma and a method due to Andrews-Newton-Zeilberger to reprove results from [5] and [24] as well as to prove new related material. 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. Combinatorial identities that were needed in [24] are proved, mostly with C. Schneider’s computer algebra package Sigma. The form of the Padé approximation of the logarithm of arbitrary order is stated as a conjecture. 1.

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.

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.