arXiv subjects
Publications and source records attributed to Minking Eie.
In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. Leveraging Zagier's formula for computing $\zeta^\star(\{2\}^p,3,\{2\}^q)$, we employ our theorems to establish a formula for computing $\zeta^\star(\{2\}^p,1,\{2\}^q)$ for any positive integers $p$ and $q$, along with other formulas of interest.
In this paper, we are going to perform the shuffle products of $Z_-(n) = \sum_{a+b=m} (-1)^{b} \zeta(\{1\}^{a},b+2)$ and $Z_+^\star(n) = \sum_{c+d=n} \zeta^{\star}(\{1\}^{c},d+2)$ with $m+n = p$. The resulted shuffle relation is a weighted sum formula given by \begin{equation*} \frac{(p+1)(p+2)}{2} \zeta(p+4) =\sum_{m+n=p} \sum_{|\boldsymbol{\alpha}|=p+3} \zeta(\alpha_{0}, \alpha_{1}, \ldots, \alpha_{m}, \alpha_{m+1}+1) \sum_{a+b+c=m} \Bigl( W_{\boldsymbol\alpha}(a,b,c) + W_{\boldsymbol\alpha}(a,b,c=0) + W_{\boldsymbol\alpha}(a=0,b,c) + W_{\boldsymbol\alpha}(a=0,b=m,c=0) \Bigr), \end{equation*} where $W_{\boldsymbol\alpha}(a,b,c) = 2^{\sigma(a+b+1)-\sigma(a)-(b+1)} (1-2^{1-\alpha_{a+b+1}}\ \ )$, with $\sigma(r) = \sum_{j=0}^{r} \alpha_{j}$.
In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height $\leq 2$ and the evaluation of $\zeta^\star(\{1\}^m,\{2\}^{n+1})$. We also give a new proof of sum formula of multiple zeta values.
In this paper, we investigate the sums of mutliple zeta(-star) values of height one: $Z_{\pm}(n)=\sum_{a+b=n} (\pm 1)^b\zeta(\{1\}^a,b+2)$, $Z_{\pm}^{\star}(n)=\sum_{a+b=n} (\pm 1)^b\zeta^{\star}(\{1\}^a,b+2)$. In particular, we prove that the weighted sum $\sum_{\substack{0\leq m\leq p\\ m: {\rm even}}} \sum_{\mid\boldsymbol{\alpha}\mid=p+3} 2^{\alpha_{m+1}\ +1}\zeta(\alpha_0,\alpha_1,\ldots,\alpha_m,\alpha_{m+1}+1) $ can be evaluated through the convolution of $Z_{-}(m)$ and $Z_{+}(n)$ with $m+n=p$.
In this paper, we investigate the Euler sums $$ G_{n+2}(p,q)=\sum_{1\leq k_1<k_2<\cdots<k_{p+1}}\frac1{k_1k_2\cdots k_pk_{p+1}^{n+2}} \sum_{1\leq\ell_1\leq\ell_2\leq\cdots\leq\ell_q\leq k_{p+1}}\frac1{\ell_1\ell_2\cdots\ell_q}. $$ We give another two representations, a reflection formula, and some other properties. Then we use these results to calculate $\zeta^\star(r+2,\{2\}^n)$, for $r=0,1,2$, as our applications.
For $k\leq n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\geq 2$. More precisely, $E(mn,k)=\sum_{|\boldsymbolα|=n}ζ(mα_1,mα_2,\ldots, mα_k)$. In this paper, we develop a formula to express $E(mn,k)$ in terms of $ζ(\{m\}^p)$ and $ζ^\star(\{m\}^q)$, $0\leq p,q\leq n$. In particular, we settle Genčev's conjecture on the evaluation of $E(4n,k)$ and also evaluate $E(mn,k)$ explicitly for small even $m\leq 8$.
In this paper, we introduce the method of adding additional factors and a parameter to multiple zeta values and prove some generalizations of the duality theorem and several relations among multiple zeta values. In particular, we are able to evaluate some special (truncated) sums in terms of Riemann zeta values of different weights.