Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values

In this paper, we will study finite multiple $T$-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author's previous conjecture that the finite Euler sum space of weight, $w$, is isomorphic to a quotient Euler sum space of weight, $w$. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important $\Q$-linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the $\Q$-span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers.


Introduction
In [10], Kaneko and Tsumura proposed a study of multiple T -values (MTVs): as level two variations of multiple zeta values (MZVs), which, in turn, were first studied by Zagier [24] and Hoffman [3] independently: where N is the set of positive integers.These series converge if and only if s 1 ≥ 2, in which case we say s is admissible.As usual, we call |s| := s 1 + • • • + s d the weight and d the depth.Note that the series becomes Riemann zeta values when d = 1.One of the most important properties of these values is that they can be expressed by iterated integrals.The main motivation to consider MTVs is that they are also equipped with the following iterated integral expressions: which provide the MTVs with a Q-algebra structure because of the shuffle product property satisfied by iterated integral multiplication (see, e.g., [27, Lemma 2.1.2(iv)]).
In addition to MTVs, many other variants of MZVs have been studied due to their important connections to a variety of objects in both mathematics and theoretical physics.For example, Yamamoto [23] defined the interpolated version of MZVs, which connects ordinary MZVs to the starred version; Hoffman [4] defined an odd variant by restricting the summation indices n j 's to odd numbers only; Xu and the author [21,22] further extended both Kaneko-Tsumura and Hoffman's versions to allow for all possible parity patterns.
On the other hand, the congruence properties of the partial sums of MZVs were first considered by Hoffman [6] and the author [26] independently.Contrary to the classical cases, only a few variants of these sums exist (see, e.g., [9,12,17]).In this paper, the author will concentrate on the finite analog of MTVs defined by (1).
Let P be the set of primes.Then by putting we can define the finite multiple zeta values (FMZVs) according to the following: Nowadays, the main motivation for studying 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., [13,14,15]).In particular, for each positive integer w ≥ 2, the element is the finite analog of ζ(w), where B n 's are the Bernoulli numbers defined by which have played very important roles in many areas of mathematical studies, such as Clifford analysis [2] and topology [11].Furthermore, the connection goes even further to their alternating versions -the Euler sums and finite Euler sums.For s 1 , . . ., s d ∈ N and σ 1 , . . ., σ d = ±1, we define the Euler sums In order to save space, if σ j = −1, then s j will be used, and if a substring, S, repeats n times in the list, then {S} n will be used.For example, the finite analog of where ♯ = * or ¡.Our primary motivation for studying alternating MTVs is to better understand this mysterious relation, f ES .One of the main results of this paper is the discovery of the linear shuffle relations among the finite alternating MTVs given by Theorem 3.2.For example, it immediately implies the highly nontrivial result in Proposition 3.16: for all d ∈ N, we have 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 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.Section 4 is devoted to presenting a few results about the alternating MTVs and providing their structures explicitly when the weight is one or two.In the last section, we 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, as obtained by Xu and the author [21,22].

Symmetric and Finite Multiple T -Values
It turns out that finite MTVs are closely related to another variant called finite MSVs.For all admissible s = (s 1 , . . ., s d ) ∈ N d , we define the finite multiple T -values (FMTVs) and the finite multiple S-values (FMSVs) as It is clear that We define the ♯-symmetric multiple T -values (SMTVs) and ♯-symmetric multiple S-values (SMSVs) as  = T S ♯ (s).The MSVs and the odd d cases can all be computed similarly and are left to the interested reader.
Hence, we expect that whenever certain relations hold on the finite side, then the same relations should hold for the symmetric version, at least modulo ζ(2), and vice versa.Sometimes, they are valid for the symmetric version even without modulo ζ(2).For example, the following reversal relations hold for both types of sums by [25, Propositions 2.8 and 2.9].For s = (s 1 , . . ., s d ), we state ← − s = (s d , . . ., s 1 ).

Linear Shuffle Relations for Finite Alternating Multiple T -Values
One of the most important tools for studying MZVs and Euler sums is to consider the double shuffle relations that are produced in two ways to express these sums: one as series (by definition) and the other as iterated integrals.This idea will play the key role in the following discovery of the linear shuffle relations for finite multiple T -values (FMTVs) and their alternating version.The linear shuffle relations for Euler sums are given in [27,Theorem 8.4.3].First, we extend MTVs and FMTVs to their alternating version.For all admissible (s, σ) ∈ N d × {±1} d (i.e., (s 1 , σ 1 ) = (1, 1)), we define the alternating multiple T -values as . This is basically the same definition we used in [20], except for a possible sign difference.If we denote the version in loc.cit.as T ′ (s; σ), 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 when treating 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 σ 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 η 1 , . . ., η d = ±1, it is then easy to evaluate the iterated integral:

By changing the indices n
where η j = σ 1 • • • σ j for all j ≥ 1 and p, q represent the conversions between the series and the integral expressions of alternating MTVs: Namely, p pushes a word used in the series definition to a word used in the integral expression, whereas q goes backward.See [20] for more details.
In order to state the linear shuffle relations among FMTVs and their alternating version, first, we quickly review the algebra setup and the corresponding results for Euler sums.Let A * 1 (resp.A * 2 ) be the Q-algebra of words on {x 0 , x 1 } (resp.{x 0 , x 1 , x −1 }) with concatenation as the product.Let A 1 j (j = 1, 2) be the sub-algebra generated by the words not ending with x 0 .Then, for each word where σ 1 = η 1 and σ j = η j /η j−1 for all j ≥ 2. Set τ (1) = 1 and , and s ∈ N, we have For alternating MTVs, we can similarly let T * 1 (resp.T * 2 ) be the Q-algebra of words on {y 0 , y 1 } (resp.{y 0 , y 1 , y −1 }) with concatenation as the product.Let T 1 j (j = 1, 2) be the sub-algebra generated by the words not ending with y 0 .Then, for each word u = y s1−1 0 y σ1 . . .
2 be the two maps defined in (8) and (9).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 (−), T A , or T (−; x) or even their partial sums, such as For all words w ∈ T 1 2 , we set T A (w) := T A,¡ (w) = T A, * q(w) .Further, set τ (1) = 1 and By taking u = ∅ and then setting w = u, we see that (ii) implies (i).By decomposing w into strings of type y s−1 0 y 1 , we see that (iii) implies (ii).So, we only need to prove (iii).For simplicity, write a = y 0 and b = y 1 for the rest of this proof.Observe that for any odd prime p, the coefficient of x p of T (s; σ; x) is nontrivial if and only if dep(s) is odd.Therefore, if the depth d of the word w is even, the coefficient of x p in T * q(bw); x is given as Hence, if dep(u) + dep(v) is odd, then, by first applying T ¡ (−; x) to the above and then extracting the coefficients of x p from both sides, we obtain from shuffling the product property of the iterated integrals.Now, the last sum is p-integral since p−j < p and j < p, and therefore, we obtain which completes the proof of (iii).
Remark 3.3.In [8], Jarossay showed that the corresponding results of Theorem 3.1 hold for SMZVs.Theorem 3.2, Conjecture 1.1 and Proposition 2.1 clearly imply that similar statements also hold true for SMTVs when the depth conditions are satisfied, as in Theorem 3.2.However, it is possible to prove this unconditionally by using the generalized Drinfeld associator Ψ 2 and considering the words of the form [27,Theorem 13.4.1].The details of this work will appear in a future paper.
We can now derive a sum formula for FMTVs.Theorem 3.4.Suppose d ∈ N is odd.For all s 1 , . . ., s d ∈ N, we have Proof.This follows immediately from the linear shuffle relation by taking s 0 = 1 and u = 1 in Theorem 3.2(iii).
The following conjecture is supported by all k ≤ 9, numerically.Conjecture 3.5.For all k ∈ N, we have Proposition 3.6.If k is odd, then for all ℓ ≤ k, we have If, in addition, we assume Conjecture 3.5 holds, then Proof.For all ℓ ≤ k, we see the linear shuffle relations Thus, by setting a ℓ = T A ({1} ℓ , 2, {1} k−ℓ ), we obtain Hence, which yields (10).Then, (11) follows immediately if we assume Conjecture 3.5.

Values at Small Depths/Weights
First, we observe that since ζ A (s) = 0 for all s ∈ N according to [27, Theorem 8.2.7] we must have where q 2 is the Fermat quotient ( 6), and β s is given in (5).Further, in depth two, according to Proposition 2.6 in our arxiv paper 2402.08160,we see that for all a, b ∈ N, if w = a + b is odd, then The depth three case is already complicated, and we do not have a general formula.This is expected since such a formula does not exist for FMZVs.In the rest of this section, we will deal with some special cases.
Next, we prove a proposition that improves a result that Tauraso and the author obtained more than a decade ago by applying the newly discovered linear shuffle relations above.Proposition 3.7.We have Proof.It immediately follows on from [19, Propositions 7.3 and 7.6] that According to the linear shuffle relations for finite Euler sums, we have which readily yields the identity which in turn quickly implies all the evaluations in the proposition.
In [27, Proposition 14.2.7], we have Thus, we get as desired.
In general, we can use linear shuffles to derive many relations from the finite Euler sums.For example, Theorem 3.13.Let FMT w be the Q-vector space generated by FMTVs of weight w.Then, we have the following generating sets for w < 7: Moreover, by using numerical computation aided by Maple (see [27,Appendix D], for the pseudo codes), we can find a generating set of FMT w for every w ≤ 13.We will list the corresponding dimensions at the end of this paper.

Homogeneous Cases
In this subsection, we will compute finite Euler sums ζ(s) when s is homogeneous, i.e., s = ({s} d ) for some s ∈ D.Then, we will consider the corresponding results for FMTVs.Proposition 3.14.Let N odd be the set of odd positive integers.For any d, s ∈ N, we have where δ s,1 is the Kronecker symbol.In particular, ζ A ({s} d ) = 0 for all even s.
One may compare the next corollary with the well-known result that ζ A ({1} d ) = 0 for all d ∈ N (see, e.g., [27,Theorem 8.5.1]).Proposition 3.17.For all d ∈ N, we have Proof.For any ℓ ∈ N, we have the relation for the regularized value (see, e.g., [7, Section 2]) According to the definition, By conducting extensive numerical experiments, we found that the following relations must be valid.Conjecture 3.18.For all odd w ∈ N, we have The conjecture holds when w = 3 according to Corollary 3.8 and Proposition 3.9.Aided by Maple, we can also rigorously prove the conjecture for w = 5 and w = 7 by using the tables of values of finite Euler sums produced by reversal, shuffle, and linear shuffle relations, and the table of values for Euler sums is available online [1].
Moreover, Conjecture 3.18 still holds true for T A ({1} w ) = T S ¡ ({1} w ) = 0 when w is even because of Propositions 3.16 and 3.17.However, for S-values, we have another conjecture.
Conjecture 3.19.For all even w ∈ N, there are rational numbers c j ∈ Q, 1 ≤ j ≤ w/2, such that Moreover, S A (j, w − j) and 1 ≤ j ≤ w/2 are Q-linearly independent, and S S ¡ (j, w − j) and 1 ≤ j ≤ w/2 are Q-linearly independent.
Note that S A (j, w − j) ∈ A when S S ¡ (j, w − j) are all real numbers.

Alternating Multiple T -Values
We now turn to the alternating version of MTVs and derive some relations among them.These values are intimately related to the colored MZVs of level 4 (i.e., multiple polylogarithms evaluated for the fourth roots of unity).We refer the interested reader to [21,22] for the fundamental results concerning these values.
Recall that for any (s, σ) ∈ N d × {±1} d , we have defined the finite alternating multiple T -values as We have seen from Theorem 3.2 in Section 3 that these values satisfy the linear shuffle relations.It is also not hard to get the reversal relations when the depth is even, as shown below.
Proof.Let p be an odd prime.Then, by changing the indices n j → p − n j , we obtain Let t j = s d+1−j , ε j = σ d+1−j , and k j = n d+1−j .Then, by changing the indices, we obtain j → d + 1 − j (since d is even) as desired.
It should be clear to the attentive reader that T -values are always intimately related to the S-values when the depth is odd because of the reversal relations.Even though we did not consider this in the above, it plays a key role in the proof of the next result.Proposition 4.2.Let q 2 (p) = (2 p−1 − 1)/2 for all p > 2.Then, we have According to [18,Theorem 3.2], we see that Since S p (1) = ζ p ( 1)/2 = −q p (2), we immediately see that S A ( 1) = −q 2 /2.By taking the reversal, we obtain as desired.
Define the finite Catalan's constant as Proposition 4.5.Let FAMT w be the vector space generated by finite alternating MTVs over Q.We have the following generating sets of FAMT w for w < 3: Proof.The w = 1 case is trivial.For w = 2, we already know T A (1, 1) = T A (2) = 0 from Theorem 3.13.Let a = y 0 , b = y 1 , and c = y −1 in the rest of the proof.For alternating values, we first have the linear shuffle relation By using complicated computation (see Proposition 4.4 of our arxiv paper 2402.08160 and notice (7)), we have the additional relation Then, from the reversal relation (15), we easily see that T A ( 1, 1) = −(−1) p ′ T A (1, 1).This completes the proof of the proposition.

Dimensions of FMT and AMT
We first need to point out that it is possible to study the alternating MTVs by converting them into colored MZVs of level 4 and then applying the setup in [17].For example, The caveat is that we need to extend our scalars to Q[i] in general.At the end of [17], we observed that dim Q FCMZ 4 w ≤ 2 w for all w ≥ 1, where FCMZ 4 w is the space spanned by all colored MVZ of level 4 and weight w over Q.By the following, we expect that the where FAM is the space spanned by all the finite multiple mixed values.Here, according to [21], the multiple mixed values mean we allow all possible even/odd combinations in the definition of such series instead of a fixed pattern, such as that which appears in MTVs and MSVs).
Conjecture-Principle-Philosophy 5.1.Let S be a set of colored MZVs (including MZVs and Euler sums) or (alternating) multiple mixed values (or their variations/analogs, such as finite, symmetric, interpolated versions, etc.).Then, the following statements should hold.
(1) Suppose all elements in S have the same weight.If they are linearly independent over Q, then they are algebraically independent over Q.
(2) If the weights of the values in S are all different, then the values are linearly independent over Q (but, of course, may not be algebraically independent over Q).
(3) If there is only one nonzero element in S, then it is transcendental over Q.
For example, we expect that ζ(n)'s are not only irrational but are also transcendental for all n ≥ 2. We also expect that q 2 and β k are transcendental for all odd k ≥ 3 and are all algebraically independent over Q Recall that MT w (resp.FMT w ) is the Q-vector space generated by MTVs (resp.finite MTVs) of weight w.Similarly, we denote by AMT (resp.FAMT) the space generated by alternating MTVs (resp.finite alternating MTVs) of weight w.From numerical computation, we conjecture the following upper bounds for the dimensions of FMT w and FAMT w .In order to compare to the classical case, we tabulate the results together in the following.The main software we used was the open source computer algebra system GP-Pari.With strong numerical support, Xu and the author conjecture that {dim Q AMT w } w≥1 form the tribonacci sequence (see [22,Conjecture 5.2]).For MTVs, Kaneko and Tsumura conjecture in [10] that, for all k ≥ 1 From numerical computation, we can formulate its finite analog as follows: Conjecture 5.2.For all k ≥ 1,

Conclusions
The author's main purpose in this paper is to study the finite and symmetric MTVs and their alternating versions, which are level two and level four variations of finite MZVs.It was found that there are many nontrivial Q-linear relations among these values, such as the reversal and the linear shuffle relations.We also numerically discovered some identities, which were proposed as conjectures.Due to the limitation of computing power, we then computed the structures of MTVs (their alternating versions) when the weight was less than 7 (resp.4) by using finite Euler sums and the relations that we discovered.Throughout the study, we were guided by the Conjecture Principle Philosophy 5.1, which provides the big picture in which the main objects of this paper lie.We plan to investigate the symmetric versions of these values more in a future paper.

Motivated by Conjecture 1 . 1 ,
we provide the following definition.

Proposition 4 . 1 (
Reversal relations of finite alternating MTVs).Let s ∈ N d for some even d ∈ N.Then,
We remark that ζ S ¡ (s) − ζ S * (s) always lies in ζ(2)ES w−2 , see [27], Exercise 8.7.Thus, it does not matter which version of the symmetric Euler sums is used in the conjecture.Problem 1.2.What is the correct generalization of [27, Theorem 6.3.5] for symmetric Euler sums?What is the correct extension of [27, Theorem 8.5.10] to finite Euler sums?