An exotic shuffle relation of $ζ(\{2\}^m)$ and $ζ(\{3,1\}^n)$

Details An exotic shuffle relation of $ζ(\{2\}^m)$ and $ζ(\{3,1\}^n)$ An exotic shuffle relation of $ζ(\{2\}^m)$ and $ζ(\{3,1\}^n)$

2007-07-23

arxiv.org/abs/0707.3244v1

Arch. Math. (Basel) 91 (5)(2008), pp. 409-415

Mathematics

Number Theory

5 pages

Scientific paper

10.1007/s00013-008-2669-x

In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: $$\zeta(\{2\}^m \sha\{3,1\}^n)={2n+m\choose m} \frac{\pi^{4n+2m}}{(2n+1)\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by WZ-method.

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