Fast Computer Algebra for Special Functions

Today, large parts of modern mathematics would be unthinkable without computers. As the first connotation of computer mathematics, one might think of numerical methods, by which for instance solutions of differential equations can be approximated. But also symbolic methods are more and more recognized as an indispensable tool for the mathematical work. Indeed, a whole lot of questions, including also quite dodgy issues, can meanwhile be settled completely and rigorously by computer programs. The proving of special function identities belongs to these questions, as well as the finding of closed form representations for sums, integrals, or power series. The processes which are applied in this context are, however, computationally very expensive. It is therefore not unusual that for a problem, although it could in principle be solved by a computer program, in practice no solution is found, just because the number of necessary elementary operations is so astronomically high that even a modern computer cannot carry them out in any reasonable amount of time. So there naturally arises the question whether alternative computational procedures could be found which can solve the same problems with fewer elementary operations. This question has been pursued in the framework of the project. We have focused on algorithms for special functions. Such algorithms have already in the past led to spectacular results, including some for which no classical (computer free) proofs are known so far. Also in the course of the project we have been able to construct some such proofs, one of them being a proof of the q-TSPP conjecture, a central problem from enumerative combinatorics. Secondly, for numerous techniques have been able to find extensive theoretical and practical improvements, without which some computer proofs would not be possible. Thirdly, we have implemented the theoretical results in open software packages, which enjoy an increasing user community. We intend to continue their development after the end of the project.