## Taiwanese Journal of Mathematics

### On Vectorized Weighted Sum Formulas of Multiple Zeta Values

#### Abstract

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

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

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874439

Digital Object Identifier
doi:10.11650/tjm.20.2016.5487

Mathematical Reviews number (MathSciNet)
MR3481383

Zentralblatt MATH identifier
1357.11079

#### Citation

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