# F50-08: Combinatorial and Algorithmic Aspects of Elliptic Hypergeometric Series

Combinatorial and algorithmic aspects of elliptic hypergeometric series will be investigated, with the overall goal to further the (relatively new) theory of elliptic hypergeometric series. While several authors have been studying elliptic hypergeometric series from a pure ``special functions point of view'', only little work has been done revealing combinatorial aspects or fundamental algorithmic aspects of these series. The two objectives on the combinatorial side of this proposal concern on one hand the use of elliptic weight functions to enumerate suitable classes of combinatorial objects, hereby obtaining elliptic generating functions. On the other hand, elliptic-commuting variables are to be further studied. On the algorithmic side it is aimed to work out an elliptic Zeilberger algorithm which can be used to determine a (theta function) recurrence for a given elliptic hypergeometric series. The next objective would be the implementation of an elliptic Petkovsek algorithm to determine closed form elliptic hypergeometric solutions of such a recurrence. Other goals concern the further study of elliptic Taylor series expansions, for single and multiple series.