Computer algebra and multiple sums

Abstract

The WZ method of Wilf and Zeilberger has become a standard tool for symbolic manipulation of  ($q$-)hypergeometric single sums and series.  Despite its tremendous success there, the problem of designing efficient computer algebra algorithms for ($q$-)hypergeometric multiple sums is still a challenge. The talk reports on two approaches that seem to be promising steps in this direction.  The first one is Wegschaider's combination of Sister Celine/WZ methodology with ideas of Verbaeten. A $q$-version will be presented in Riese's talk. The second one is Schneider's extension of Karr's summation theory. Karr's approach works over certain difference field extensions and can be viewed as an analogue to Risch's integration algorithm.