Taiwanese Journal of Mathematics

On Vectorized Weighted Sum Formulas of Multiple Zeta Values

Chan-Liang Chung and Yao Lin Ong

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In this paper, we introduce the vectorization of shuffle products of two sums of multiple zeta values, which generalizes the weighted sum formula obtained by Ohno and Zudilin. Some interesting weighted sum formulas of vectorized type are obtained, such as\begin{align*}  &\quad \sum_{\substack{\boldsymbol{\alpha}+\boldsymbol{\beta}=\boldsymbol{k}    \\ |\boldsymbol{\alpha}|: \textrm{even}}}    M(\boldsymbol{\alpha}) M(\boldsymbol{\beta}) \sum_{|\boldsymbol{c}|=|\boldsymbol{k}|+r+3} 2^{c_{|\boldsymbol{\alpha}|+1}}    \zeta(c_0, c_1, \ldots, c_{|\boldsymbol{\alpha}|}, \ldots,    c_{|\boldsymbol{k}|+1}+1) \\  &= \frac{1}{2} (2|\boldsymbol{k}|+r+5) M(\boldsymbol{k})    \zeta(|\boldsymbol{k}|+r+4),\end{align*}where $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{k}$ are $n$-tuples of nonnegative integers with $|\boldsymbol{k}| = k_1 + k_2 + \cdots + k_n$ even; $M(\boldsymbol{u})$ is the multinomial coefficient defined by $\binom{u_1 + u_2 + \cdots + u_n}{u_1, u_2, \ldots, u_n}$ with the value $\frac{|\boldsymbol{u}|!}{u_1! u_2! \cdots u_n!}$; and $r$ is a nonnegative integer. Moreover, some newly developed combinatorial identities of vectorized types involving multinomial coefficients by extending the shuffle products of two sums of multiple zeta values in their vectorizations are given as well.

Article information

Taiwanese J. Math., Volume 20, Number 2 (2016), 243-261.

First available in Project Euclid: 1 July 2017

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Zentralblatt MATH identifier

Primary: 40A25: Approximation to limiting values (summation of series, etc.) {For the Euler-Maclaurin summation formula, see 65B15} 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M06: $\zeta (s)$ and $L(s, \chi)$ 33E20: Other functions defined by series and integrals

multiple zeta value shuffle relation sum formula weighted sum formula


Chung, Chan-Liang; Ong, Yao Lin. On Vectorized Weighted Sum Formulas of Multiple Zeta Values. Taiwanese J. Math. 20 (2016), no. 2, 243--261. doi:10.11650/tjm.20.2016.5487. https://projecteuclid.org/euclid.twjm/1498874439

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