Another proofs of Zagier's formula for multiple zeta values and Murakami's formula for multiple $t$-values
Jinmin Yu, Shaofang Hong
Abstract
Let $l\ge 1$ be an integer. For any multiple index $\mathbf{s}=(s_1,s_2,\cdots,s_l)\in\mathbb{Z}_{\geq 1}^l$ with $s_l>1$, the multiple zeta value (MZV for short) is defined by \begin{align*} ζ(s_1,s_2,\cdots,s_l):=\sum_{1\leq k_1<k_2<\cdots<k_l} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_l^{s_l}} \end{align*} and the multiple $t$-value is defined by \begin{align*} t(s_1,s_2,...,s_l):=\sum_{1\leq k_1<k_2<...<k_l} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_l-1)^{s_l}}, \end{align*} where if the index is empty, then we define the value $t(\emptyset):=1$. We denote by $\{a_1,\cdots,a_k\}^d$ the sequence formed by repeating the sequence $\{a_1,\cdots,a_k\}$ exactly $d$ times. Let $H(r,s)=ζ(\{2\}^r,3,\{2\}^s)$ and $T(r,s):=t(\{2\}^r,3,\{2\}^s)$. Zagier's formula for the multiple zeta values $H(r,s)$ was an important and key ingredient in the proof of Hoffman's conjecture. In this paper, with the help of the Lei-Yu-Hong expressions for $H(r,s)$ and $T(r,s)$ as well as Lupu's identity about rational zeta series involving Riemann zeta values $ζ(2n)$ and by establishing some identities about binomial coefficients and a result about Kronecker symbol and arithmetic functions, we present another proofs of Zagier's formula stating that for any nonnegative integers $r$ and $s$, \begin{align*} H(r,s)=2\sum_{k=1}^{r+s+1}(-1)^k\Big[\binom{2k}{2r+2}-\Big(1-\frac{1}{2^{2k}}\Big) \binom{2k}{2s+1}\Big]ζ(2k+1)ζ(\{2\}^{r+s+1-k}), \end{align*} and Murakami's formula for the multiple $t$-values $T(r,s)$ asserting that \begin{align*} T(r,s)=\sum_{k=1}^{r+s+1}(-1)^{k-1} \Big[\binom{2k}{2r+1}+\binom{2k}{2s+1}\Big(1-\frac{1}{2^{2k}}\Big)\Big] \frac{1}{2^{2k}}ζ(2k+1) t(\{2\}^{r+s+1-k}). \end{align*}