In [10] Kaneko and Tsumura proposed to study the multiple T-values (MTVs) as level two variations of multiple zeta values which in turn were first studied by Zagier [26] and Hoffman [3] independently: where $\mathbb{N}$ is the set of positive integers. These series converge if and only if $s_1\ge 2$ in which case we say s is admissible. As usual, we call $\left|\mathit{s}\right|:=s_1+\cdots +s_d$ the weight and d the depth. The main motivation to consider MTVs is that they have the following iterated integral expressions which equips the MTVs with a $\mathbb{Q}$-algebra structure because of the shuffle product property satisfied by the iterated integral multiplication (see, e.g., [25, Lemma 2.1.2(iv)]).

Besides MTVs, many other variants of multiple zeta values have been studied due to their important connections to a varieties of objects in both mathematics and theoretical physics (see, e.g., [2,6,10,22]). On the other hand, the congruence properties of the partial sums of MZVs were first considered by Hoffman [5] and the author [24] independently. Contrary to the classical cases, only a few variants of these sums exist (see, e.g., [9,16,23]). In this paper, we will concentrate on the finite analog of MTVs defined by (1).

Let $\mathcal{P}$ be the set of primes and put Then we can define the finite multiple zeta values (FMZVs) by the following:Nowadays, the main motivation to study FMZVs is to understand a deep conjecture proposed by Kaneko and Zagier around 2014 (see Conjecture 1.1 below for a generalization). Although this conjecture is far from being proved many parallel results have been shown to hold for both MZVs and FMZVs simultaneously (see, e.g., [12,13,14]). In particular, for each positive integer $w\ge 2$, the element is the finite analog of $\zeta \left(w\right)$, where $B_n$’s are the Bernoulli numbers defined by

And the connection goes even further to their alternating versions—the Euler sums and finite Euler sums. For $s_1,\cdots ,s_d\in \mathbb{N}$ and ${\sigma }_1,\cdots ,{\sigma }_d=\pm 1$, we define the Euler sumsTo save space, if ${\sigma }_j=-1$ then ${\overline{s}}_j$ will be used and if a substring S repeats n times in the list then ${\left\{S\right\}}^n$ will be used. For example, the finite analog of $-\zeta \left(\overline{1}\right)=-\zeta (1;-1)=log2$ is the Fermat quotient Put $sgn\left(\overline{s}\right)=-1$ and $|\overline{s}|=s$ if $s\in \mathbb{N}$. For $s_1,\dots ,s_d\in \mathbb{D}:=\mathbb{N}\cup \overline{\mathbb{N}}$ we can define the finite Euler sums by

In [25, Conjecture 8.6.9] we extended Kakeko–Zagier conjecture to the setting of the Euler sums. For $s=(s_1,\dots ,s_d)\in {\mathbb{D}}^d$, define the symmetric version of the alternating Euler sums where ${\zeta }_{♯}$ ($♯=\ast$ or ⧢) are regularized values (see [25, Proposition 13.3.8]). They are called ♯-regularized symmetric Euler sums. If $s\in {\mathbb{N}}^d$ then they are called ♯-regularized symmetric multiple zeta values (SMZVs).

We remark that ${\zeta }_{⧢}^{\mathcal{S}}\left(\mathit{s}\right)-{\zeta }_{\ast }^{\mathcal{S}}\left(\mathit{s}\right)$ always lies in $\zeta \left(2\right){\mathsf{ES}}_{w-2}$, see [25, Exercise 8.7]. Thus it does not matter which version of symmetric Euler sums is used in the conjecture.

Our primary motivation to study finite (alternating) MTVs is to better understand this mysterious relation $f_{\mathsf{ES}}$. We now briefly describe the content of this paper. We will start the next section by defining finite MTVs and symmetric MTVs, which can be shown to appear on the two sides of Conjecture 1.1, respectively. The most useful property of MTVs is that they have the iterated integral expressions (3) satisfying the shuffle multiplication. This leads us to the discovery of the linear shuffle relations for the finite MTVs (and their alternating version) in section 3 and some interesting applications of these relations. In the last section, we will consider both the finite MTVs and their alternating version by computing the dimension of the weight w piece for $w<9$ and then compare these data to their Archimedean counterparts obtained by Xu and the author [19,20].

It turns out that the finite MTVs are closely related to another variant called finite MSVs. For all admissible $\mathit{s}=(s_1,\cdots ,s_d)\in {\mathbb{N}}^d$, we define the finite multiple T-values (FMTVs) and the finite multiple S-values (FMSVs) by It is clear that

Motivated by Conjecture 1.1, we provide the following definition.

Hence, we expect that whenever certain relation hold on the finite side then the same relations should hold for the symmetric version, at least modulo $\zeta \left(2\right)$, and vice versa. Sometimes, they are valid for the symmetric version even without modulo $\zeta \left(2\right)$. For example, the following reversal relations hold for both types of sums (see [23, Proposition 2.8 and 2.9]). For $\mathit{s}=(s_1,\cdots ,s_d)$ we put $\overleftarrow{\mathit{s}}=(s_d,\cdots ,s_1)$.

One of the most important tools to study MZVs and Euler sums is to consider the double shuffle relations which are produced by two ways to express these sums: one as series by definition, the other by iterated integrals. This idea will play the key role in the following discovery of the linear shuffle relations for FMTVs and their alternating version.

The linear shuffle relations for Euler sums are given by [25, Theorem 8.4.3]. First, we extend MTVs and FMTVs to their alternating version. For all admissible $(\mathit{s},\mathit{\sigma })\in {\mathbb{N}}^d\times {\{\pm 1\}}^d$ (i.e. $(s_1,{\sigma }_1)\ne (1,1)$), we define the alternating multiple T-values by This is basically the same definition we used in [20,21] except for a possible sign difference. If we denote by $T^{\prime }(\mathit{s};\mathit{\sigma })$ the version in loc. cit., then We changed to our new convention in this paper because of the significant simplification in this special case. However, the old convention is still superior to treat the general alternating multiple mixed values. Similar to the convention for Euler sums, we will save space by putting a bar on top of $s_j$ if ${\sigma }_j=-1$. For example, In order to study the alternating MTVs, it is to our advantage to consider the alternating multiple T-functions of one variable as follows. For any real number x, define In the non-alternating case, this function is the A-function (up to a power of 2) used by Kaneko and Tsumura in [10]. For all ${\eta }_1,\cdots ,{\eta }_d=\pm 1$, it is then easy to evaluate the iterated integral Let By the change of indices $n_j=2k_j+2k_{j+1}+\cdots +2k_d+d-j+1$ we immediately get where ${\eta }_j={\sigma }_1\cdots {\sigma }_j$ for all $j\ge 1$ and $\mathbf{p},\mathbf{q}$ represent the conversions between the series and the integral expressions of alternating MTVs: Namely, $\mathbf{p}$ pushes a word used in the series definition to a word used in the integral expression while $\mathbf{q}$ goes backwards. See [21] for more details.

To state the linear shuffle relations among FMTVs and their alternating version, we first quickly review the algebra setup and the corresponding results for Euler sums. Let ${\mathfrak{A}}_1^{\ast }$ (resp. ${\mathfrak{A}}_2^{\ast }$) be the $\mathbb{Q}$-algebra of words on $\{{\mathtt{x}}_0,{\mathtt{x}}_1\}$ (resp. $\{{\mathtt{x}}_0,{\mathtt{x}}_1,{\mathtt{x}}_{-1}\}$) with concatenation as the product. Let ${\mathfrak{A}}_j^1$ ($j=1,2$) be the subalgebra generated by the words not ending with ${\mathtt{x}}_0$. Then for each word $\mathbf{u}={\mathtt{x}}_0^{s_1-1}{\mathtt{x}}_{{\eta }_1}\cdots {\mathtt{x}}_0^{s_d-1}{\mathtt{x}}_{{\eta }_d}\in {\mathfrak{A}}_2^1$, we define where ${\sigma }_1={\eta }_1$ and ${\sigma }_j={\eta }_j/{\eta }_{j-1}$ for all $j\ge 2$. Set $\tau \left(1\right)=1$ and

For alternating MTVs, we can similarly let ${\mathfrak{T}}_1^{\ast }$ (resp. ${\mathfrak{T}}_2^{\ast }$) be the $\mathbb{Q}$-algebra of words on $\{{\mathtt{y}}_0,{\mathtt{y}}_1\}$ (resp. $\{{\mathtt{y}}_0,{\mathtt{y}}_1,{\mathtt{y}}_{-1}\}$) with concatenation as the product. Let ${\mathfrak{T}}_j^1$ ($j=1,2$) be the subalgebra generated by the words not ending with ${\mathtt{y}}_0$. Then for each word $\mathbf{u}={\mathtt{y}}_0^{s_1-1}{\mathtt{y}}_{{\sigma }_1}\cdots {\mathtt{y}}_0^{s_d-1}{\mathtt{y}}_{{\sigma }_d}\in {\mathfrak{A}}_2^1$, let $\mathbf{p},\mathbf{q}:{\mathfrak{A}}_2^1\to {\mathfrak{A}}_2^1$ be the two maps defined by (18) and (19). Then we can extend the definition of alternating MTVs and their corresponding one-variable functions to the word level: where $F(-)$ can be either $T(-)$, or $T_{\mathcal{A}}$, or $T(-;x)$ or ⧢ even their partial sums such as

For all words $\mathbf{w}\in {\mathfrak{T}}_2^1$ we set $T_{\mathcal{A}}\left(\mathbf{w}\right):=T_{\mathcal{A},⧢}\left(\mathbf{w}\right)=T_{\mathcal{A},\ast }\left(\mathbf{q}\left(\mathbf{w}\right)\right).$ Further, set $\tau \left(1\right)=1$ and

We can now derive a sum formula for FMTVs.

The following conjecture is supported by all $k\le 9$ numerically.