1 Introduction

The systematic study of multiple zeta values (MZVs) started in the early 1990s with the work of Hoffman [Reference Hoffman4] and Zagier [Reference Zagier, Joseph, Mignot, Murat, Prum and Rentschler13]. Let $r\ge 1$ be an integer. For any multiple index $\mathbf {s}=(s_1,s_2,\ldots ,s_r)\in \mathbb {Z}_{\geq 1}^r$ with $s_r > 1$ , the MZV is defined by

$$ \begin{align*} \zeta(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{k_1^{s_1}k_2^{s_2}\cdots k_r^{s_r}}. \end{align*} $$

The weight of the MZV is defined to be the number $|\mathbf {s}|:=s_1+s_2+\cdots +s_r$ and its depth is defined by $l(\mathbf {s}):=r$ . For brevity, we let $\{a_1,a_2,\ldots ,a_k\}^d$ denote the sequence formed by repeating the sequence $\{a_1,a_2,\ldots ,a_k\}$ exactly d times. The simplest precise evaluations of MZVs are

$$ \begin{align*} \zeta(\{2\}^d)=\frac{\pi^{2d}}{(2d+1)!}. \end{align*} $$

A central problem for MZVs is to investigate the algebraic structure of the $\mathbb {Q}$ -vector space $\mathfrak {Z}_k$ spanned by MZVs of weight k. In this direction, Hoffman [Reference Hoffman5] proposed a conjecture that states that every MZV of weight k can be expressed as a $\mathbb {Q}$ -linear combination of those MZVs whose entries are only $2$ ’s and $3$ ’s. This conjecture was proved by Brown [Reference Brown2] using a motivic method. In particular, Brown showed that $H(a,b):=\zeta (\{2\}^a,3,\{2\}^b)$ can be expressed as a $\mathbb {Q}$ -linear combination of the products $\pi ^{2m}\zeta (2n+1)$ with $m+n=a+b+1$ . In 2012, Zagier [Reference Zagier14] found a surprisingly elegant and explicit formula as follows.

Theorem 1.1 (Zagier [Reference Zagier14])

For nonnegative integers a and b,

(1.1) $$ \begin{align} H(a,b)=2\sum_{k=1}^{a+b+1}(-1)^kc^{(k)}_{a,b}\zeta(2k+1) \zeta(\{2\}^{a+b+1-k}), \end{align} $$

where

$$ \begin{align*}c^{(k)}_{a,b}:=\binom{2k}{2a+2}-\bigg(1-\frac{1}{2^{2k}}\bigg)\binom{2k}{2b+1}.\end{align*} $$

The proof of Theorem 1.1 is given by computing the associated generating functions of both sides in a closed form and then showing they are entire functions of exponential growth that agree at sufficiently many points to force their equality. Zagier’s proof was simplified by Li [Reference Li8]. In 2017, K. Hessami Pilehrood and T. Hessami Pilehrood [Reference Hessami Pilehrood and Hessami Pilehrood3] gave an alternative proof of Theorem 1.1.

In a similar way, Hoffman [Reference Hoffman6] defined the multiple t-values (‘odd variant’ of MZVs). For any multiple index $\mathbf {s}=(s_1,s_2,\dots ,s_r)\in \mathbb {Z}_{\geq 1}^r$ , the multiple t-value (MtV) is defined by

$$ \begin{align*} t(s_1,s_2,\ldots,s_r):=\sum_{1\leq k_1<k_2<\cdots<k_r} \frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}\cdots(2k_r-1)^{s_r}}. \end{align*} $$

If the index is empty, then we define the value $t(\emptyset ):=1$ . From [Reference Murty, Sinha, Friedberg, Bump, Goldfeld and Hoffstein11, Reference Zhao15] and analogous to the simplest multiple zeta value,

$$ \begin{align*} t(\mathop{\underbrace{2,2,\ldots,2}}_{n})=\frac{\pi^{2n}}{2^{2n}(2n)!}. \end{align*} $$

By the method of generating functions, Murakami [Reference Murakami10] obtained an equivalent version of Theorem 1.1. Define

$$ \begin{align*}\widetilde{t}(k_1,k_2,\ldots,k_r):=2^{|k|}t(k_1,k_2,\ldots,k_r)\end{align*} $$

and set

$$ \begin{align*}K(a,b):=\widetilde{t}(\{2\}^a,3,\{2\}^b), \quad K(n)=\widetilde{t}(\{2\}^n)\end{align*} $$

for any $a,b,n\in \mathbb {N}_{\geq 0}$ . Then, for all integers $a,b\geq 0$ ,

(1.2) $$ \begin{align} K(a,b)=\sum_{k=1}^{a+b+1}(-1)^{k-1}d^{(k)}_{a,b}K(a+b-k+1)\zeta(2k+1), \end{align} $$

where

$$ \begin{align*}d^{(k)}_{a,b}:=\binom{2k}{2a+1}+\bigg(1-\frac{1}{2^{2k}}\bigg)\binom{2k}{2b+1}.\end{align*} $$

Moreover, let $T(a,b):=t(\{2\}^a,3,\{2\}^b)$ . Then, (1.2) is equivalent to

(1.3) $$ \begin{align} T(a,b)=\sum_{k=1}^{r+s+1}(-1)^{k-1}d^{(k)}_{a,b}\frac{1}{2^{2k}}\zeta(2k+1) t(\{2\}^{a+b-k+1}). \end{align} $$

In a different approach, Lupu [Reference Lupu9] provided elementary proofs of Zagier’s formula for the special case $b=0$ , and also established a Zagier-type formula for multiple t-values when $b=0$ . For the general case $b>0$ , Lupu [Reference Lupu9] proposed the following two conjectures.

Conjecture 1.2. For nonnegative integers a and b,

(1.4) $$ \begin{align} H(a,b)=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}. \end{align} $$

Conjecture 1.3. For nonnegative integers a and b,

(1.5) $$ \begin{align} &T(a,b) \notag \\ &\quad=\frac{-2}{(2a+1)!}\bigg(\frac{\pi}{2}\bigg)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align} $$

Conjectures 1.2 and 1.3 are shown in [Reference Lupu9] to be true when $b=0$ . However, for the general case $b>0$ , these two conjectures are still open.

In 2026, Lai et al. [Reference Lai, Lupu and Orr7] gave elementary and direct proofs of the results stated above for both $H(a, b)$ and $T(a, b)$ . They first provided an integral expression for $H(a, b)$ and then showed that this integral expression is equal to Zagier’s result. The same proof method was applied to $T(a, b)$ .

In this paper, our main goal is to investigate Conjectures 1.2 and 1.3. We will prove that both conjectures of Lupu are true. By employing the integral expressions presented in [Reference Lai, Lupu and Orr7] for $H(a,b)$ and $T(a,b)$ , and using the properties of the Beta and Gamma functions, we show the main results of this paper.

Theorem 1.4. For nonnegative integers a and b,

$$ \begin{align*} H(a,b)=\frac{-4\pi^{2a+2b+2}}{(2a+2)!}\sum_{n=0}^{\infty} \frac{\zeta(2n)}{(2n+2a+2)(2n+2a+3)\cdots(2n+2a+2b+3)2^{2n}}. \end{align*} $$

Theorem 1.5. For nonnegative integers a and b,

$$ \begin{align*} &T(a,b) \\ &\quad=\frac{-2}{(2a+1)!}\bigg(\frac{\pi}{2}\bigg)^{2a+2b+2} \sum_{n=0}^{\infty}\frac{\zeta(2n)}{(2n+2a+1)(2n+2a+2)\cdots(2n+2a+2b+2)2^{2n}}. \end{align*} $$

This paper constitutes the first step in giving another proof for both Zagier’s and Murakami’s formulae (see the subsequent paper by Yu and Hong [Reference Yu and Hong12]). Our new rational zeta series approach could lead to other evaluations for different families of MZVs and MtVs.

We organise this paper as follows. In Section 2, we supply several preliminary lemmas that are needed in the proofs of our main results. Sections 3 and 4 are devoted to the proofs of Theorems 1.4 and 1.5, respectively.