## Sum formula for finite multiple zeta values.(English)Zbl 1329.11093

M. Kaneko [“Finite multiple zeta values $$\mod p$$ and relations among multiple zeta values”, Sūrikaisekikenkyūsho Kōkyūroku 1813, 27–31 (2010)] conjectured that $\sum_{k_1+\cdots+ k_n=k,\,k_1\geq 2}\zeta_{\mathcal A}(k_1,\dots,k_n)=\left(1+(-1)^n\binom{k-1}{n-1}\right)\frac{B_{p-k}}{k},$ and $\sum_{k_1+\cdots+ k_n=k,\,k_1\geq 2}\zeta_{\mathcal A}^{\star}(k_1,\dots,k_n)=\left((-1)^n+\binom{k-1}{n-1}\right)\frac{B_{p-k}}{k},$ where $$\zeta_{\mathcal A}$$ and $$\zeta_{\mathcal A}^{\star}$$ are finite multiple zeta(-star) values.
In this paper, the authors prove these conjectures and extend the results for a wider class of summation domain. These are the finite versions of the corresponding well-known summation formulas of MZV theory.

### MSC:

 11M32 Multiple Dirichlet series and zeta functions and multizeta values

### Keywords:

finite multiple zeta value; sum formula
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### References:

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