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F50-09: Computer Algebra for Nested Sums and Products

Principal Investigator: Carsten Schneider

This project part aims at simplification and solving methods that are relevant for combinatorial and related problems, like, e.g., plane partitions, rhombus tilings, the statistical analysis of combinatorial objects, or the analysis of algorithms. In particular, the project part deals with symbolic simplification of combinatorial formulas given in terms of complicated nested multiple sums and with solving of linear recurrence relations. The underlying algorithms will be based on an improved summation theory in the context of difference fields and rings. Combining these new techniques, one will obtain a fully developed toolbox that is able to find alternative representations for answers of combinatorial problems in terms of special functions that are expressible in terms of indefinite nested sums and products. In addition, the project part will elaborate algorithms that will extract additional information from these representations. E.g., new constructive tools will be developed to prove algebraic independence of indefinite nested sums or to calculate asymptotic expansions of such objects. The ultimate goal will be to apply the proposed computer algebra algorithms to non-trivial problems within the SFB. In this regard, the tools will be tuned and adapted by the needs and challenges of  the emerging combinatorial problems.

Carsten Schneider