Towards a Symbolic Summation Theory for Unspecified Sequences
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.
We would like to thank Christian Krattenthaler for inspiring discussions. Special thanks go to Bill Chen and his collaborators at the Center of Applied Mathematics at the Tianjin University for overwhelming hospitality in the endspurt phase of writing up this paper. We are especially grateful for all the valuable and detailed suggestions of the referee that improved substantially the quality of this article.
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