2012/05/31 by Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura
Mathematics · #Action (physics) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Parity (physics) #Riemann zeta function #Root (linguistics) #Type (biology) #math.NT #msc:11M06 #msc:11M32
paper · pdf · doi:10.7169/facm/2014.51.1.3
published as Funct. Approx. Comment. Math. 51 (2014), 43-76 · 27 pages
arxiv created 2012/08/29 · openalex publication_date 2014/09/01 · arxiv updated 2016/04/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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-function of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type Cr. 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 Cr. 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 Br analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.