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Abstract: We present summation algorithms for a certain class of summands, which we call the "level-2-sequences". They include (q-)hypergeometric sequences, P-finite sequences, and many sequences involving special functions from mathematical physics. The algorithms of Sister Celine and Kurt Wegschaider are generalized for computing (multiple) sums of level-2-sequences. In the single sum case, F.Chyzak's extension of Zeilberger's algorithm can be adapted to level-2-sequences in a straightforward way. All algorithms presented are implemented in Mathematica.
Abstract: The talk gives an overview of the symbolic summation capabilities and recent developments in Maple. Feedback is very much appreciated.
Abstract: We present a new algorithm for computing the dispersion set of two integer polynomials. The approach is similar to that of Man and Wright (1994), but only requires p-adic factorization of the input polynomials. We present a rigourous analysis and implementation experiments comparing this method with previous approaches. We also describe extensions to computing dispersion of polynomials over number fields and of polynomials with parameters. This is joint work with Jurgen Gerhard, Arne Storjohann and Eugene Zima.
Abstract: Sigma is a summation package, implemented in the computer algebra system Mathematica, that enables to discover and prove nested multisum identities. Based on Karr's difference field theory (1981) this package allows to find all solutions of parameterized linear difference equations in a very general difference field setting, so called Π-Σ-fields. With a refined version of this difference field machinery indefinite multisums can be simplified by minimizing the depth of nested sum-quantifiers. In addition, Sigma provides several algorithms in order to discover closed form evaluations of definite nested multisums. Here one first tries to compute a recurrence for a given definite sum by applying Zeilberger's creative telescoping idea in the difference field setting. Second one attempts to solve this recurrence in terms of d'Alembertian solutions, a subclass of Liouvillian solutions. As it turns out, our indefinite summation algorithm plays a major role in order to simplify those solutions further. Combining these simplified solutions one finally may find a closed form evaluation of a definite multisum. All these aspects will be illustrated by various examples.
Abstract: The mutiple gamma function, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in this function has been revived. In this talk I will discuss some theoretical aspects of the multiple gamma function and its applications to summation of series and infinite products.
Abstract: Let F(n,k) be a rational function in two variables. For positive integers n, let R(n) be the sum of F(n,k) for k from 0 to n. We present a method, that we conjecture to be a complete algorithm, to compute R(n) if R(n) is a rational function in n.
Abstract: Gosper's algorithm takes as input a univariate hypergeometric term tk and computes, if possible, a rational function R(k) such that sk = R(k) tk satisfies the recurrence sk+1 - sk = tk. However, R(k) can have integer poles and hence this recurrence need not be valid for all integer k. Similarly, Zeilberger's algorithm takes as input a bivariate hypergeometric term F(n,k) and computes, if possible, a rational function R(n,k), and a linear recurrence operator L with polynomial coefficients (depending only on n), such that G(n,k) = R(n,k) F(n,k) satisfies the recurrence G(n,k+1) - G(n,k) = L F(n,k). Again, R(n,k) can have integer singularities and hence this recurrence need not be valid for all integer n and k. In this talk, we discuss this phenomenon and its influence on the correctness of the final result.
Abstract: The computation of definite integrals presents one with a variety of choices. There are various methods such as Newton-Leibniz or Slater's convolution method. There are issues such as whether to split or merge sums, how to search for singularities on the path of integration, when to issue conditional results, how to assess (possibly conditional) convergence, and more. These various considerations moreover interact with one another in ways that are not necessarily comfortable. This talk will discuss these various issues, with examples. I will describe some of the successful strategies and some of the open areas (read: not terribly successful to date) for further work. The focus is not technical in regard to the specifics of integration methods, but rather is on illustrating some of the problems one faces in constructing a practical implementation. It is based on my work on Mathematica's definite integration over the past year and a half.