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Creative Telescoping on Multiple Sums

Abstract

We showcase a collection of practical strategies to deal with a problem arising from an analysis of integral estimators derived via quasi-Monte Carlo methods. The problem reduces to a triple binomial sum, thereby enabling us to open up the holonomic toolkit, which contains tools such as creative telescoping that can be used to deduce a recurrence satisfied by the sum. While applying these techniques, a host of issues arose that partly needed to be resolved by hand. In other words, no creative telescoping implementation currently exists that can resolve all these issues automatically. Thus, we felt the need to compile the different strategies we tried and the difficulties that we encountered along the way. In particular, we highlight the necessity of the certificate in these computations and how its complexity can greatly influence the computation time.

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Acknowledgements

We would like to thank the organizers of CASC 2020 for providing an occasion and opportunity to give a talk about the work on which this article is based. We were encouraged by the positive feedback from the audience, which motivated this post-proceedings contribution. We would especially like to acknowledge Pierre Lairez for pointing us to his paper [2] and for demonstrating how to make the computation in Sect. 3.8 with his binomial sums Maple package. We thank the reviewers for their careful reading, which helped us improve this manuscript greatly, particularly the second reviewer who pointed out some related literature and provided insightful criticism. We also express our appreciation to Hao Du and Ali Uncu for their support and helpful commentary.

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Correspondence to Elaine Wong.

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Both authors were supported by the Austrian Science Fund (FWF): F5011-N15.

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Koutschan, C., Wong, E. Creative Telescoping on Multiple Sums. Math.Comput.Sci. 15, 483–498 (2021). https://doi.org/10.1007/s11786-021-00514-3

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Keywords

  • Symbolic summation
  • Creative telescoping
  • Holonomic function
  • Hypergeometric series

Mathematics Subject Classification

  • 33F10
  • 68W30
  • 33C05
  • 39A13