Summation//Integration and Algebraic Relations

This project deals with the problem of symbolic summation, i.e., with the problem of finding closed form representations for (finite) sums. The overall goal is the development of algorithms that are able to simplify a given sum expression -- or that deliver a proof that the given sum expression cannot be simplified any further (in a certain, well defined sense). Algorithms for symbolic summation already exist for several types of sum expressions. For the case of a hypergeometric summand, there are algorithms which have been developed starting from the late seventies with the discovery of Gosper`s algorithm. These algorithms are nowadays included in all major computer algebra systems and may hence be considered as classical. For some more sophisticated classes of summation problems, there do exist algorithms, too. In the proposed research, we focus on summation problems to which no summation algorithm is currently applicable. We are especially interested in summation problems where the summand is expressed in terms of sequences that obey nontrivial algebraic relations. This is because the presence of such algebraic relations cause serious problems in known summation algorithms. On the other hand, recent work of the proposer has led to progress concerning the discovery of algebraic relations, that forms a promising basis for the proposed research. We are not only interested in summation algorithms but also in summation methods based on heuristic approaches, where resorting to heuristics approaches brings some advantages, e.g., the advantage of covering a wider range of summation problems. We are also interested in integration problems that can be transformed into summation problems to which summation algorithms are applicable. The expected outcome of the project is interesting from a algorithmic/computational viewpoint. It is also interesting with respect to applications. We expect that the summation methods developed in the frame of this project will be able to solve summation and integration problems arising, e.g., in the theory of special functions or in combinatorics, which are of independent interest.

This project deals with the problem of symbolic summation, i.e., with the problem of finding closed form representations for (finite) sums. The overall goal is the development of algorithms that are able to simplify a given sum expression - or that deliver a proof that the given sum expression cannot be simplified any further (in a certain, well defined sense). Algorithms for symbolic summation already exist for several types of sum expressions. For the case of a hypergeometric summand, there are algorithms which have been developed starting from the late seventies with the discovery of Gosper`s algorithm. These algorithms are nowadays included in all major computer algebra systems and may hence be considered as classical. For some more sophisticated classes of summation problems, there do exist algorithms, too. In the proposed research, we focus on summation problems to which no summation algorithm is currently applicable. We are especially interested in summation problems where the summand is expressed in terms of sequences that obey nontrivial algebraic relations. This is because the presence of such algebraic relations cause serious problems in known summation algorithms. On the other hand, recent work of the proposer has led to progress concerning the discovery of algebraic relations, that forms a promising basis for the proposed research. We are not only interested in summation algorithms but also in summation methods based on heuristic approaches, where resorting to heuristics approaches brings some advantages, e.g., the advantage of covering a wider range of summation problems. We are also interested in integration problems that can be transformed into summation problems to which summation algorithms are applicable. The expected outcome of the project is interesting from a algorithmic/computational viewpoint. It is also interesting with respect to applications. We expect that the summation methods developed in the frame of this project will be able to solve summation and integration problems arising, e.g., in the theory of special functions or in combinatorics, which are of independent interest.