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

TL;DR

This paper presents a new, concise proof of an exotic shuffle relation involving multiple zeta values, extending previous results and utilizing combinatorial identities verified by the WZ-method.

Contribution

The paper offers a shorter proof of a general shuffle relation for multiple zeta values, building on prior work and applying the WZ-method for verification.

Findings

Established a new proof for the shuffle relation involving multiple zeta values.

Extended known results to a more general case using combinatorial identities.

Validated identities through the WZ-method.

Abstract

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.