Differentiation and Integration of Indefinite Summations with Respect to Indexed Variables — Some Rules and Applications

In this paper we present some rules for the differentiation and integration of expressions involving indefinite summations with respect to indexed variables which have not yet been taken into account of current computer algebra systems. These rules, together with several others, have been implemented in MACSYMA and MAPLE as a toolkit for manipulating indefinite summations. We discuss some implementation issues and report our experiments with a set of typical examples. The present work is motivated by our investigation in the computer-aided analysis and derivation of artificial neural systems. The application of our rules to this subject is briefly explained.