Another proofs of Zagier's formula for multiple zeta values and Murakami's formula for multiple $t$-values

TL;DR

This paper provides new proofs for Zagier's formula for multiple zeta values and Murakami's formula for multiple t-values, using identities involving binomial coefficients, zeta series, and arithmetic functions.

Contribution

The paper introduces alternative proofs for two important formulas related to multiple zeta and t-values, expanding the theoretical understanding of these special functions.

Findings

New proofs of Zagier's formula for H(r,s)

New proofs of Murakami's formula for T(r,s)

Utilization of binomial identities and zeta series in proofs

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*} \zeta(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 $H(r,s)=\zeta(\{2\}^r,3,\{2\}^s)$ 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 $\zeta(2n)$ 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]\zeta(2k+1)\zeta(\{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}}\zeta(2k+1) t(\{2\}^{r+s+1-k}). \end{align*}