Proofs of Lupu's conjectures for multiple zeta values and multiple $t$-values
Wenzhong Lei, Jinmin Yu, Shaofang Hong
Abstract
Let $r\ge 1$ be an integer. For any multiple index $\mathbf{s}=(s_1,s_2,\cdots,s_r) \in\mathbb{Z}_{\geq 1}^r$ with $s_r>1$, the multiple zeta value (MZV for short) is defined by \begin{align*} ζ(s_1,s_2,\cdots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}} \end{align*} and the multiple $t$-value is defined by \begin{align*} t(s_1,s_2,...,s_r):=\sum_{1\leq k_1<k_2<...<k_r} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}}, \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 $H(a,b)=ζ(\{2\}^a,3,\{2\}^b)$ and $T(a,b):=t(\{2\}^a,3,\{2\}^b)$. In this paper, by using the Lai-Lupu-Orr integral expressions for $H(a,b)$ and $T(a,b)$ and the properties of Beta function and Gamma function, we show that for any nonnegative integers $a$ and $b$, we have \begin{align*} H(a,b):=\frac{-4π^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{ζ(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}} \end{align*} and \begin{align*} T(a,b)=\frac{-2}{(2a+1)!}\left(\fracπ{2}\right)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{ζ(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align*} This confirms two conjectures of Lupu proposed in [C. Lupu, Another look at Zagier's formula for multiple zeta values involving Hoffman elements, Math. Z. 301 (2022), 3127-3140].