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Symbolic Summation with Single-Nested Sum Extensions

C. Schneider


We present a streamlined version of Karr's summation algorithm. Karr's original approach constructively decides the telescoping problem in PiSigma-fields, a very general class of difference fields that can describe rational terms of arbitrarily nested indefinite sums and products. More generally, our new algorithm can decide constructively if there exists a so called single-nested PiSigma-extension over a given PiSigma-field in which the telescoping problem for f can be solved in terms that are not more nested than f itself. This allows to eliminate an indefinite sum over f by expressing it in terms of additional sums that are not more nested than f. Moreover, our refined algorithm contributes to definite summation: it can decide constructively if the creative telescoping problem for a fixed order can be solved in single-nested PiSigma-extensions that are less nested than the definite sum itself.

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