Discovering and Proving Infinite Binomial Sums Identities

Abstract

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of π or log (2). In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.

KEYWORDS:

• binomial sums
• multiple zeta values
• cyclotomic harmonic polylogarithms

2000 AMS Subject Classification:

• 05A10
• 68W30
• 11M32
• 33F05

Acknowledgments

I would like to thank C. Schneider and F. Chyzak for calling my attention to [Sun 14]. Additionally, I want to thank C. Schneider for useful discussions.

Funding

Supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).

Notes

¹ The package HarmonicSums can be downloaded at http://www.risc.jku.at/research/combinat/software/HarmonicSums.

² The full list of the base identities that we found is available at http://arxiv.org/abs/1507.01703.

³ The full list of the base identities that we found is available at http://arxiv.org/abs/1507.01703.