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Identity involving symmetric sums of regularized multiple zeta-star values. (English) Zbl 1457.11118
Multiple zeta star values are real numbers defined by nested sums $\zeta^{\star}(k_1,\ldots,k_r)=\sum_{0< m_1\leq m_2\leq \ldots \leq m_r}\frac{1}{m_1^{k_1}m_2^{k_2}\ldots m_r^{k_r}},$ where $$k_1,\ldots,k_r \geq 1$$ are positive integers with $$k_r\geq 2$$. In the case where $$k_r=1$$, the above series diverges but can be regularized in two different ways to yield polynomials $$\zeta^{\star}_{\ast}(k_1,\ldots,k_r;T), \zeta^{\star}_{\mathrm{sh}}(k_1,\ldots,k_r;T) \in \mathbb R[T]$$ [M. Kaneko and S. Yamamoto, Sel. Math., New Ser. 24, No. 3, 2499–2521 (2018; Zbl 1435.11114)]. These two regularizations differ inasmuch as that the $$\zeta^{\star}_{\ast}$$ satisfy the stuffle (or harmonic) product formula while the $$\zeta^{\star}_{\mathrm{sh}}$$ satisfy the shuffle product formula.
The main result of the present paper is a closed formula for the symmetric sum $\sum_{\sigma\in S_r}\zeta^{\star}_{\mathrm{sh}}(k_\sigma(1),\ldots,k_\sigma(r);T)$ of shuffle-regularized multiple zeta star values of depth $$r$$, where $$S_r$$ denotes the symmetric group on $$r$$ letters. The analogous formula in the stuffle case has been proved in [M. E. Hoffman, Kyushu J. Math. 69, No. 2, 345–366 (2015; Zbl 1382.11066)]. The proof given in the present paper is similar to earlier work of the same author [Pac. J. Math. 286, No. 2, 307–359 (2017; Zbl 1396.11109)], and uses Bell polynomials.

##### MSC:
 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11B73 Bell and Stirling numbers
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