Central Binomial Sums, Multiple Clausen Values and Zeta Values

Borwein, J. M.; Broadhurst, D. J.; Kamnitzer, J.

Central Binomial Sums, Multiple Clausen Values and Zeta Values

Authors

Borwein, J. M.

Broadhurst, D. J.

Kamnitzer, J.

Description

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.
17 pages, LaTeX, with use of amsmath and amssymb packages, to appear in Journal of Experimental Mathematics