A study on multiple zeta values from the viewpoint of zeta-functions of root systems
A study on multiple zeta values from the viewpoint of zeta-functions of root systems
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DOI：
10.7169/facm/2014.51.1.3
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发表时间：
2012-05
期刊：
影响因子：
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通讯作者：
Y. Komori;Kohji Matsumoto;Hirofumi Tsumura
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文献类型：
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作者：
Y. Komori;Kohji Matsumoto;Hirofumi Tsumura
我们从前人研究过的与根系相关的ζ函数的角度研究了多重ζ值（mzv）。实际上，Euler-Zagier型的$r$-ple ζ函数可以看作是与某一类型为$C_r$的子根相关联的ζ函数。因此，通过Weyl群的作用，我们可以发现mzv的新方面，这意味着Hoffman和Zagier给出的众所周知的mzv公式与Witten与上述类型为$C_r$的子根系统相关的体积公式是一致的。同时，我们还证明了一些新的公式，特别是包含双zeta值和三zeta值的宇称结果。作为另一个重要的应用，我们给出了限制和公式的某些改进，给出了任意深度$r$的mzv之间的限制和公式，这些公式以前只在深度$2,3,4$的情况下才知道。进一步地，我们可以类比地考虑一类$B_r$的子根，给出Hoffman-Zagier公式、宇称结果和限制和公式的相关类比。
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.