Symbolic summation with single-nested sum extensions

  • Authors:

  • Carsten Schneider

  • Affiliations:

  • J. Kepler University Linz, Linz, Austria

  • Venue:
  • ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
  • Year:
  • 2004

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Abstract

We present a streamlined and refined version of Karr's summation algorithm. Karr's original approach constructively decides the telescoping problem in ΠΣ-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 ΠΣ-extension over a given ΠΣ-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 Σ*-extensions that are less nested than the definite sum itself.