# Chiral catalysis of nuclear fusion in molecules

Dmitri E. Kharzeev<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794-3800, USA  Department of Physics, Brookhaven National Laboratory,
Upton, New York 11973-5000, USA  Jake Levitt<EMAIL_ADDRESS>Cortex Fusion Systems, Inc., New York, NY 10128, USA

###### Abstract

At low energies, nuclear fusion is strongly affected by electron screening of
the Coulomb repulsion among the fusing nuclei. It may thus be possible to
catalyze nuclear fusion in molecules (i.e., to fuse specific nuclei in situ)
through quantum control of electron wave functions in intense laser fields.
The circularly polarized (chiral) laser field can effectively squeeze the
electron wave functions, greatly enhancing the screening in the spatial region
relevant for the fusion process. We estimate the corresponding fusion
probabilities, and find that the proposed chiral catalysis of nuclear fusion
in molecules may be observable, potentially with important practical
applications.

_Introduction_ Harnessing the energy of nuclear fusion is one of the grand
challenges of modern science. Under normal conditions, the Coulomb repulsion
barrier keeps the nuclei apart and prevents them from fusing. Two main ways to
overcome this repulsion are vigorously pursued at present: magnetic
confinement and inertial confinement, see [1, 2] for reviews. In both cases,
the tunneling through the Coulomb barrier is enhanced by creating a plasma,
and thus increasing the kinetic energy of the colliding nuclei.

In this letter, we investigate whether the fusion can be made to occur in situ
(i.e., inside molecules), without first creating a plasma. We argue that this
may be possible when the electrons in a molecule are controlled by a
circularly polarized ultrafast laser. Our idea stems from the fact that at low
energies the fusion probability is exponentially sensitive to the screening of
nuclear Coulomb potential. This screening in molecules is provided by the
constituent electrons, and is normally not strong enough to allow fusion –
otherwise, our world would not be stable!

However, the electron wave functions in atoms and molecules can be efficiently
controlled by lasers. In particular, in the field of a high-frequency,
circularly polarized laser, the electrons can be made to move along toroidal
orbits of radius that is significantly smaller than the Bohr radius of
innermost electrons. As a result, the Coulomb screening should become greatly
enhanced, and this may lead to observable fusion rates, potentially suitable
for a fusion reactor. The circularly polarized (chiral) nature of laser field
is crucial for this mechanism of fusion, and we will thus refer to it as the
chiral catalysis.

Our proposal is related to two important earlier ideas. The first one is the
muon-catalyzed fusion ($\mu\textrm{CF}$) proposed long time ago by Frank [3],
Sakharov [4] and Zel’dovich [5]. In $\mu$CF, the Coulomb screening is enhanced
due to the presence of muons that are about 200 times heavier than electrons
and thus move much closer to the nuclei than electrons. The $\mu$CF was
discovered experimentally by Alvarez and collaborators in 1957 [6]. The
practical use of $\mu$CF is however challenging due to the necessity to first
produce a sufficient flux of muons, that moreover have a relatively short
lifetime of about $2\ \mu$s; see [7, 8] for reviews.

The second, much less known, idea was put forward by Saha, Markmann and
Batista [9] who proposed that the probability of tunneling through the Coulomb
barrier can be enhanced by modifying the nuclear wave packet by laser pulses.
Unfortunately, the effect of laser radiation on nuclei at visible and even
soft X-ray wavelengths and moderate intensities is quite weak due to the
smallness of the nuclear Compton wavelength (or, equivalently, due to mismatch
of energies between the photons and nuclei).

However, as we will discuss in detail below, lasers do have a strong effect on
electron wave packets, and can be used to effectively turn electrons into
heavier particles moving close to the nuclei and providing the needed
screening of the Coulomb potential.

_The effect of Coulomb screening on nuclear fusion_ Nuclear fusion is governed
by the competition between the short-range attractive nuclear force and the
long-range Coulomb repulsion among the fusing nuclei. The importance of
Coulomb repulsion at a given kinetic center-of-mass energy $E$ is quantified
by the Sommerfeld parameter

$\eta(E)=\frac{Z_{1}Z_{2}e^{2}}{v(E)},$ (1)

where $v(E)=\sqrt{2E/\mu}$ is the relative velocity of the nuclei with charges
$Z_{1}$ and $Z_{2}$ and masses $M_{1}$ and $M_{2}$;
$\mu=M_{1}M_{2}/(M_{1}+M_{2})$ is the reduced mass. For low energies (say, in
KeV range), $v\ll 1$, and the Sommerfeld parameter is very large, $\eta\gg 1$
– this means that Coulomb effects are extremely important.

The fusion cross section can be written as [10]

$\sigma(E)=\frac{S(E)}{E}\exp{[-2\pi\eta(E)]},$ (2)

where $S(E)$ is the “astrophysical S-factor” that accounts for nuclear
interactions and (in the absence of near-threshold resonances) varies slowly
with energy. The exponential in (2) accounts for the suppression due to
tunneling through the repulsive Coulomb potential

$U(r)=\frac{Z_{1}Z_{2}e^{2}}{r},$ (3)

evaluated in semiclassical WKB approximation, as originally derived by Gamow
[11]. The WKB Euclidean action is evaluated between the classical turning
point $r_{t}=Z_{1}Z_{2}e^{2}/E$ determined by the condition $E=U(r_{t})$ and
the nuclear-scale distance $r_{c}$ at which the compound nucleus is formed.

If the fusing nuclei are part of atoms or molecules, the Coulomb potential (3)
is screened by electrons. The appropriate treatment of the screening effects
depends on the initial energy $E$. At low energies, when the relative velocity
of the nuclei $v(E)$ is much smaller than the Bohr velocity of bound electrons
$v_{e}\sim Z\alpha c$ (with $\alpha=e^{2}/4\pi$), $v(E)\ll v_{e}$, the
electron cloud can quickly adjust to the motion of the nuclei, and the
adiabatic Born-Oppenheimer approximation is appropriate.

In this adiabatic approximation, the effect of Coulomb screening can be
properly accounted for by the screened potential, for which we will assume a
simple Debye form

$U_{s}(r)=\frac{Z_{1}Z_{2}e^{2}}{r}\exp\left(-\frac{r}{r_{D}}\right),$ (4)

where $r_{D}$ is the screening radius. The effect of screening on the
tunneling probability can be estimated following the approach originally
proposed by Salpeter [12] and developed in [13, 14, 15]; see [16] for a recent
review. Namely, the height of the Coulomb barrier near the distance $r_{c}$
where the compound nucleus is formed can be estimated from

$U_{s}(r)\simeq\frac{Z_{1}Z_{2}e^{2}}{r}\left(1-\frac{r}{r_{D}}\right)\equiv
U(r)-U_{0},$ (5)

where we have used the Taylor expansion near $r=0$, justified by $r_{c}\ll
r_{D}$. The height of the Coulomb barrier thus gets reduced by the amount

$U_{0}=\frac{Z_{1}Z_{2}e^{2}}{r_{D}},$ (6)

and the corresponding increase of the tunneling probability can be estimated
by an effective increase of energy, $E\to E+U_{0}$. The ratio of penetration
probabilities with and without the screening is thus given by

$R_{s}(E)\equiv\frac{\exp{[-2\pi\eta(E+U_{0})]}}{\exp{[-2\pi\eta(E)]}}.$ (7)

The quantity (6) can be interpreted as the difference between the energy of
the compound atom and the sum of energies of atoms before fusion [12].
Numerically, for $d+^{2}H$ we estimate from (6) $U_{0}\simeq 24$ eV by putting
$r_{D}$ equal to the Bohr radius $r_{B}=(m_{e}\ c\ \alpha)^{-1}\simeq 0.53\
{\AA}$ (we use the natural system of units with $\hbar=1$ and
$e^{2}=\alpha\simeq 1/137$). Extracting $U_{0}$ from the energy difference of
compound atom and individual deuterium atom and hydrogen molecule (assuming an
equally weighted combination of the lowest-energy gerade and ungerade wave
functions for the electron) yields $U_{0}\simeq 20.4$ eV [17], a reasonably
close value.

For heavier atoms, one can estimate (6) by putting the screening radius
$r_{D}$ equal to the radius of the innermost electron orbit, $r_{D}\simeq
r_{B}/Z_{i}$, where $r_{B}=(m_{e}c\alpha)^{-1}\simeq 0.53\ {\AA}$ is the Bohr
radius (we use the natural system of units with $\hbar=1$) [13]. If $Z_{2}\gg
Z_{1}$, we need to consider the effect of electron screening only around the
nucleus with charge $Z_{2}$ and can neglect the effect of screening around the
other nucleus. For example, for the $p+^{16}O\to^{17}F$ fusion process with
$Z_{2}=8$ and $Z_{1}=1$, we estimate $r_{D}\simeq r_{B}/Z_{2}$ and
$U_{0}\simeq e^{2}Z_{2}^{2}/r_{B}\simeq 1.54$ keV.

The fusion cross sections are usually studied in experiment at energies $E\gg
U_{0}$, when the factor (7) can be approximately rewritten as

$R_{s}(E)\simeq\exp{\left[\pi\eta(E)\frac{U_{0}}{E}\right]},\hskip
8.5359ptE\gg U_{0}.$ (8)

This factor leads to a dramatic $\sim\exp({\rm const}/E^{3/2})$ exponential
enhancement of the fusion cross section at low energies confirmed by
experiment, see e.g. [18].

In this letter we will mostly be interested in the ultra-low energy limit of
$E\ll U_{0}$. Performing Taylor expansion of the exponent in (7) to leading
order in $E/U_{0}$, we get

$R_{s}(E)\simeq\exp{\left[2\pi\eta(E)\right]}\times\exp{\left[-2\pi
Z_{1}Z_{2}e^{2}\sqrt{\frac{\mu}{2U_{0}}}\right]},$ (9) $E\ll U_{0}.$

We can multiply (2) by this factor to obtain the fusion cross section at
ultra-low energies with the account of Coulomb screening effect:

$\sigma_{low}(E)\simeq\frac{S(E)}{E}\exp{\left[-2\pi
Z_{1}Z_{2}e^{2}\sqrt{\frac{\mu}{2U_{0}}}\right]}.$ (10)

We can observe that the Coulomb screening leads to the replacement of the
vanishingly small at $E\to 0$ Gamow factor $\exp({-{\rm const}/\sqrt{E}})$ by
a small but finite, energy-independent “screened penetration factor”

$P^{low}_{s}\equiv\exp{\left[-2\pi
Z_{1}Z_{2}e^{2}\sqrt{\frac{\mu}{2U_{0}}}\right]}.$ (11)

The value of this factor for the $p+{\rm{}^{16}O}\to{\rm{}^{17}F}$ fusion
process that we considered above is $\exp{(-147.2)}\simeq 10^{-64}$. If
instead we estimated the Coulomb suppression using the traditional unscreened
Gamow formula $\exp{(-2\pi\eta(E))}$ at, say, $E=10$ eV characteristic for
intra-molecular motion, we would get a much smaller factor of
$\exp{(-300)}\simeq 10^{-130}$.

_Fusion rates in molecules_ Let us check whether this difference can lead to
any observable effects in molecules. Consider first the $p+d$ fusion in
hydrogen deuteride molecule ${\rm HD}$. Using the screening radius
$r_{D}=r_{B}$, we get $U_{0}\simeq 24\ {\rm eV}$, corresponding to the
screened penetration factor (11)

$P^{low}_{s}({\rm HD})\sim 10^{-102}.$ (12)

The stretch vibration frequency of the hydrogen deuteride molecule is about
$\Omega({\rm HD})\simeq 4,340\ {\rm cm^{-1}}\simeq 1\times 10^{14}\ {\rm
s^{-1}}$ [19]. Assuming that the $p+d\to{\rm{}^{3}He}$ fusion process occurs
in water every time the Coulomb barrier is penetrated, we estimate the rate of
fusion processes in hydrogen deuteride as

$\lambda_{f}({\rm HD})\simeq P^{low}_{s}({\rm HD})\times\Omega({\rm HD})\simeq
10^{-88}\ {\rm s^{-1}}.$ (13)

A fusion rate per unit volume per unit time is obtained by multiplying this
rate by the density of molecules. Assuming $n({\rm HD})\simeq 10^{24}\ {\rm
cm^{-3}}$,

$f({\rm HD})=n({\rm HD})\times\lambda_{f}({\rm HD})\simeq 10^{-64}\ {\rm
cm^{-3}s^{-1}},$ (14)

which is clearly unobservable. Achieving the fusion rate $f({\rm HD})\sim{\rm
cm^{-3}\ s^{-1}}$ would require decreasing the screening radius by a factor of
18. This estimate is consistent with the results of a more sophisticated
analysis based on the R-matrix method [14].

Let us now perform an estimate for the $p+{\rm{}^{16}O}\to{\rm{}^{17}F}$
fusion in water. The symmetric $O-H$ stretch frequency in water molecule is
$\Omega({\rm H_{2}O})\simeq 3,657\ {\rm cm^{-1}}\simeq 1\times 10^{14}\ {\rm
s^{-1}}$ [20]. The rate of fusion processes in water can thus be estimated as

$\lambda_{f}({\rm H_{2}O})\simeq P^{low}_{s}({\rm H_{2}O})\times\Omega({\rm
H_{2}O})\simeq 10^{-50}\ {\rm s^{-1}}.$ (15)

corresponding to the characteristic lifetime of a gaseous water molecule

$T_{f}({\rm H_{2}O})=\lambda_{f}^{-1}({\rm H_{2}O})\simeq 3\times 10^{42}\
{\rm years}.$ (16)

A fusion rate per unit volume per unit time for the density of water $n({\rm
H_{2}O})\simeq 3\ 10^{22}\ {\rm cm^{-3}}$ is

$f({\rm H_{2}O})=n({\rm H_{2}O})\lambda_{f}({\rm H_{2}O})\simeq 3\times
10^{-28}{\rm cm^{-3}s^{-1}}.$ (17)

This estimate explains the stability of water w.r.t. nuclear fusion. Is there
a way to make the fusion rate observable? According to our result (11), the
key to enhancing the fusion rate is to increase $U_{0}$, i.e., to decrease the
screening radius $r_{D}$ below its value $r_{D}({\rm H_{2}O})\simeq
r_{B}/8\simeq 0.07\ {\AA}$. The rate $f({\rm H_{2}O})\sim{\rm cm^{-3}\
s^{-1}}$ can be achieved by decreasing the screening radius by a factor of 5.

_Chiral catalysis of fusion by a circularly polarized laser_ Is there a way of
squeezing the electron cloud by a factor of ten or a hundred? This may be
possible by putting atoms in a laser field. Assuming that the laser frequency
$\omega$ is large compared to the frequency of atomic transitions, the
effective potential of the electron-nucleus interaction can be obtained by
averaging over the fast oscillations of the laser field.

For linearly polarized laser field, this leads to so-called “atomic dichotomy”
[21], when the spherical electron distribution splits into two lobes separated
by the “classical excursion parameter”

$r_{C}=\frac{eE}{m_{e}\omega^{2}},$ (18)

where $E$ is the peak laser electric field. This parameter represents the
distance traveled by the electron under the influence of Lorentz force
$m_{e}a=-eE$ in the oscillating electric field $E$, $at^{2}/2\sim
a/\omega^{2}\sim eE/(m_{e}\omega^{2})$.

However, for our purposes a much more interesting case is the circularly
polarized laser field. In this case, the electron distribution becomes ring-
like [22, 23], with symmetry axis oriented along the propagation of light, and
going through the nucleus. The radius of the ring is equal to $r_{C}$, which
corresponds to the circular motion of the electron at angular frequency
$\omega$ that is phase-locked to the electric field of the laser. Indeed, for
a circularly polarized laser field with frequency $\omega$ the classical
equation of motion of the electron under the influence of the Lorentz force
$F(t)=-eE(t)$ results in a circular motion in the transverse $(x,y)$ plane
(assuming that the light propagates along the $z$ axis) with angular frequency
$\omega$ and the radius $r_{C}$.

In the presence of the Coulomb potential $U=-Ze^{2}/r$, the effective
electron-nucleus potential can be obtained by using Kapitza method [24], where
the slow motion of the electron in Coulomb field and fast motion in the laser
field are separated. The resulting effective potential in $(1+1)$ dimensions
is then [25, 26]

$U_{eff}=U+\frac{<F^{2}>}{4m_{e}\omega^{2}}+\frac{<F^{2}>}{4m_{e}\omega^{4}}\frac{d^{2}U}{dx^{2}},$
(19)

where $<F^{2}>$ is the Lorentz force averaged over the fast motion. The second
and third terms on the r.h.s. of (19) represent the ponderomotive energy, i.e.
the time-averaged energy of electron’s harmonic motion. After a
straightforward generalization to 3D case, the effective electron-nucleus
potential (19) becomes [22, 26]

$U_{eff}(r)=-\frac{Ze^{2}}{r}+\frac{r_{C}^{2}}{r^{3}}\ P_{2}(\cos{\theta}),$
(20)

where $\theta$ is the angle between the radius vector and the direction in
which the laser field propagates, and
$P_{2}(\cos\theta)=\frac{1}{2}(3\cos^{2}\theta-1)$ is the Legendre polynomial.
At $\theta=\pi/2$, this potential exhibits strong attraction in addition to
the Coulomb one, and leads to an electron cloud squeezed along the directions
perpendicular to the direction of propagating light. This potential is
mathematically equivalent to the gravitational potential experienced by a
satellite orbiting a prolate planet [26].

_Required laser parameters_ What are the requirements on a laser needed to
achieve $r_{C}$ that is smaller than the typical screening radius by an order
of magnitude? For our treatment to hold, the laser frequency has to be larger
than atomic transition frequencies, and the laser intensity has to be large
enough to justify the semiclassical approximation. Combining these
requirements with the condition for the smallness of $r_{C}$ in comparison to
the radius of the innermost electron orbit in a molecule, we find that an
ideal laser frequency is in the soft X-ray range.

It may however be possible to approach the desired regime even with the
tabletop UV lasers. For example, a UV Lyman-alpha laser with wavelength of 122
nm has the frequency of $\omega\simeq 2.4\ 10^{15}$ Hz. Laser electric field
$E$ is related to the laser intensity $I$ by $I=\epsilon_{0}cE^{2}/2$, where
$\epsilon_{0}$ is the permittivity of free space. This leads to
$I(W/cm^{2})=1.33\ 10^{-3}[E(V/cm)]^{2}$. Assuming $I=10^{9}\ W/cm^{2}$, we
get $E=10^{6}\ V/cm$, and $r_{C}\sim 0.02\ {\AA}$ which is about five times
smaller than the screening radius in the water molecule. According to the
estimates given above, this should lead to an observable fusion rate.

Unfortunately, it is not clear whether the semiclassical treatment that we
used would hold at relatively low intensities. To answer this question, we
plan to investigate the corresponding Floquet dynamics in the water molecule
numerically.

Let us note however that the available tabletop laser systems utilizing higher
harmonic generation produce wavelengths down to 1 nm. It can thus be possible
to reduce the screening radius $r_{C}\sim\sqrt{I}/\omega^{2}$ using a suitable
combination of intensity $I$ and frequency $\omega$ within the domain of
applicability of our treatment.

_Summary_ Chiral control of electron screening in molecules by circularly
polarized lasers may open a new pathway towards nuclear fusion. However
further theoretical and experimental studies are necessary to establish
whether the corresponding fusion rates are observable and suitable for
practical applications. _Intellectual property_ The concepts, systems,
apparati, and techniques articulated herein are protected by U.S. Patent No.
63/596,122.

## Acknowledgement

The work of D.K. was supported in part by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, Grants Nos. DE-FG88ER41450 and
DE-SC0012704.

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

Jianqiang<EMAIL_ADDRESS>

Department of Mathematics, The Bishop’s School, La Jolla, CA 92037, USA

Abstract. 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 $\mathbb{Q}$-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 $\mathbb{Q}$-span of finite (alternating) MTVs for some small weights
by rigorously using the newly discovered relations, numerically aided by
computers.

Keywords. (finite) Euler sums; symmetric Euler sums; (finite) multiple
$T$-values; symmetric multiple $T$-values; alternating multiple $T$-values.

2020 Mathematics Subject Classification. 11M32; 11B68; 68W30.

##  1 Introduction

In [10], Kaneko and Tsumura proposed a study of _multiple $T$-values_ (MTVs):

$\displaystyle
T({\boldsymbol{\sl{s}}}):=\sum_{\begin{subarray}{c}n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{1}{n_{j}^{s_{j}}},\quad{\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d},$
(1)

as level two variations of _multiple zeta values_ (MZVs), which, in turn, were
first studied by Zagier [24] and Hoffman [3] independently:

$\zeta({\boldsymbol{\sl{s}}}):=\sum_{n_{1}>\dots>n_{d}>0}\prod_{j=1}^{d}\frac{1}{n_{j}^{s_{j}}},\quad{\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d},$
(2)

where $\mathbb{N}$ is the set of positive integers. These series converge if
and only if $s_{1}\geq 2$, in which case we say ${\boldsymbol{\sl{s}}}$ is
admissible. As usual, we call $|{\boldsymbol{\sl{s}}}|:=s_{1}+\cdots+s_{d}$
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:

$T({\boldsymbol{\sl{s}}})=\int_{0}^{1}\left(\frac{dt}{t}\right)^{s_{1}-1}\frac{dt}{1-t^{2}}\cdots\left(\frac{dt}{t}\right)^{s_{d}-1}\frac{dt}{1-t^{2}}$
(3)

which provide the MTVs with a $\mathbb{Q}$-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 $\mathcal{P}$ be the set of primes. Then by putting

$\mathcal{A}:=\prod_{p\in\mathcal{P}}(\mathbb{Z}/p\mathbb{Z})\bigg{/}\bigoplus_{p\in\mathcal{P}}(\mathbb{Z}/p\mathbb{Z}),$
(4)

we can define the _finite multiple zeta values_ (FMZVs) according to the
following:

$\zeta_{\mathcal{A}}({\boldsymbol{\sl{s}}}):=\bigg{(}\sum_{p>n_{1}>\dots>n_{d}>0}\prod_{j=1}^{d}\frac{1}{n_{j}^{s_{j}}}\pmod{p}\bigg{)}_{p\in\mathcal{P}}\in\mathcal{A}.$

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 $w\geq 2$, the element

$\beta_{w}:=\Big{(}\frac{B_{p-w}}{w}\Big{)}_{w<p\in\mathcal{P}}\in\mathcal{A}$
(5)

is the finite analog of $\zeta(w)$, where $B_{n}$’s are the Bernoulli numbers
defined by

$\frac{t}{e^{t}-1}=\sum_{n\geq 0}B_{n}\frac{t^{n}}{n!},$

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 $s_{1},\dots,s_{d}\in\mathbb{N}$ and
${\sigma}_{1},\dots,{\sigma}_{d}=\pm 1$, we define the _Euler sums_

$\zeta(s_{1},\dots,s_{d};{\sigma}_{1},\dots,{\sigma}_{d}):=\sum_{n_{1}>\cdots>n_{d}>0}\;\prod_{j=1}^{d}\frac{{\sigma}_{j}^{n_{j}}}{n_{j}^{s_{j}}}.$

In order to 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
$\\{S\\}^{n}$ will be used. For example, the finite analog of
$-\zeta(\bar{1})=-\zeta(1;-1)=\log 2$ is the Fermat quotient

${\texttt{q}}_{2}:=\Big{(}\frac{2^{p-1}-1}{p}\pmod{p}\Big{)}_{3\leq
p\in\mathcal{P}}\in\mathcal{A}.$ (6)

Write $\operatorname*{sgn}(\bar{s})=-1$ and $|\bar{s}|=s$ if $s\in\mathbb{N}$.
For $s_{1},\ldots,s_{d}\in\mathbb{D}:=\mathbb{N}\cup\bar{\mathbb{N}}$, we can
define the finite _Euler sums_ as

$\zeta_{\mathcal{A}}({\boldsymbol{\sl{s}}}):=\bigg{(}\sum_{p>n_{1}>\dots>n_{d}>0}\prod_{j=1}^{d}\frac{\operatorname*{sgn}(s_{j})^{n_{j}}}{n_{j}^{|s_{j}|}}\pmod{p}\bigg{)}_{p\in\mathcal{P}}\in\mathcal{A}.$

In [27, Conjecture 8.6.9], we extended the Kakeko–Zagier conjecture to the
setting of the Euler sums. For
${\boldsymbol{\sl{s}}}=(s_{1},\ldots,s_{d})\in\mathbb{D}^{d}$, define the
symmetric version of the alternating Euler sums

$\displaystyle\zeta_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}}):=$
$\displaystyle\sum_{i=0}^{d}\bigg{(}\prod_{j=1}^{i}(-1)^{|s_{j}|}\operatorname*{sgn}(s_{j})\bigg{)}\zeta_{\sharp}(s_{i},\dots,s_{1})\zeta_{\sharp}(s_{i+1},\dots,s_{d})$

where $\zeta_{\sharp}$ ($\sharp=\ast$ or $\shuffle$) are regularized values
(see [27, Proposition 13.3.8]). They are called $\sharp$-regularized
_symmetric Euler sums_. If ${\boldsymbol{\sl{s}}}\in\mathbb{N}^{d}$, then they
are called $\sharp$-regularized _symmetric multiple zeta values_ (SMZVs).

######  Conjecture 1.1 (cf. [27, Conjecture 8.6.9]).

For any $w\in\mathbb{N}$, let $\mathsf{FES}_{w}$ (resp. $\mathsf{ES}_{w}$) be
the $\mathbb{Q}$-vector space generated by all finite Euler sums (resp. Euler
sums) of weight $w$. Then, there is an isomorphism:

$\displaystyle f_{\mathsf{ES}}:\mathsf{FES}_{w}$
$\displaystyle\longrightarrow\frac{\mathsf{ES}_{w}}{\zeta(2)\mathsf{ES}_{w-2}},$
$\displaystyle\zeta_{\mathcal{A}}({\boldsymbol{\sl{s}}})$
$\displaystyle\longmapsto\zeta_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}}),$

where $\sharp=\ast$ or $\shuffle$.

We remark that
$\zeta_{\shuffle}^{\mathcal{S}}({\boldsymbol{\sl{s}}})-\zeta_{\ast}^{\mathcal{S}}({\boldsymbol{\sl{s}}})$
always lies in $\zeta(2)\mathsf{ES}_{w-2}$, 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?

Our primary motivation for studying alternating MTVs is to better understand
this mysterious relation, $f_{\mathsf{ES}}$. 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 $d\in\mathbb{N}$, we
have

$T_{\mathcal{A}}(\\{1\\}^{2d})=0.$

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].

##  2 Symmetric and Finite Multiple $T$-Values

It turns out that finite MTVs are closely related to another variant called
finite MSVs. For all admissible
${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$, we define the
_finite multiple $T$-values_ (FMTVs) and the _finite multiple $S$-values_
(FMSVs) as

$\displaystyle
F_{\mathcal{A}}({\boldsymbol{\sl{s}}}):=\bigg{(}\sum_{\begin{subarray}{c}p>n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\ 2)}\ \text{if }F=T,\\\ n_{j}\equiv d-j{\
(\rm{mod}\ 2)}\ \text{if
}F=S\end{subarray}}\prod_{j=1}^{d}\frac{1}{n_{j}^{s_{j}}}\pmod{p}\bigg{)}_{p\in\mathcal{P}}\in\mathcal{A}.$

It is clear that

$F_{\mathcal{A}}({\boldsymbol{\sl{s}}})=\frac{1}{2^{d}}\sum_{{\sigma}_{1},\dots,{\sigma}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\ 2|d-j\ \text{if $F=T$}\\\
2\nmid d-j\ \text{if
$F=S$}\end{subarray}}{\sigma}_{j}\bigg{)}\zeta_{\mathcal{A}}({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}}).$

Motivated by Conjecture 1.1, we provide the following definition.

######  Definition 2.1.

Let $d\in\mathbb{N}$ and
${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$. Let $F=S$ or
$T$. We define the _$\sharp$ -regularized MTVs_ ($\sharp=\ast$ or $\shuffle$)
and _MSVs_ as

$\displaystyle F_{\sharp}({\boldsymbol{\sl{s}}}):=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{{\sigma}_{1},\dots,{\sigma}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\ 2|d-j\ \text{if $F=T$}\\\
2\nmid d-j\ \text{if
$F=S$}\end{subarray}}{\sigma}_{j}\bigg{)}\zeta_{\sharp}({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}})\quad(\text{$F=T$
or $S$}).$

We define the _$\sharp$ -symmetric multiple $T$-values_ (SMTVs) and _$\sharp$
-symmetric multiple $S$-values_ (SMSVs) as

$F_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}}):=\left\\{\begin{array}[]{ll}\displaystyle\sum_{i=0}^{d}\Big{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}\Big{)}F_{\sharp}(s_{i},\dots,s_{1})F_{\sharp}(s_{i+1},\dots,s_{d}),&\quad\hbox{if
$d$ is even;}\\\
\displaystyle\sum_{i=0}^{d}\Big{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}\Big{)}\widetilde{F}_{\sharp}(s_{i},\dots,s_{1})F_{\sharp}(s_{i+1},\dots,s_{d}),&\quad\hbox{if
$d$ is odd,}\end{array}\right.$

where $\widetilde{F}=S+T-F$ and we set, as usual, $\prod_{\ell=1}^{0}=1$.

######  Proposition 2.1.

Suppose $f_{\mathsf{ES}}$ is defined as per Conjecture 1.1. Let $\sharp=\ast$
or $\shuffle$. Then, for all
${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})\in\mathbb{N}^{d}$, we have
$f_{\mathsf{ES}}T_{\mathcal{A}}({\boldsymbol{\sl{s}}})=T_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}})$
and
$f_{\mathsf{ES}}S_{\mathcal{A}}({\boldsymbol{\sl{s}}})=S_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}})$
modulo $\zeta(2)$.

###### Proof.

Suppose $d$ is even and ${\boldsymbol{\sl{s}}}\in\mathbb{N}^{d}$. Then, modulo
$\zeta(2)$

$\displaystyle f_{\mathsf{ES}}T_{\mathcal{A}}({\boldsymbol{\sl{s}}})=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\ j\equiv d{\ (\rm{mod}\
2)}\end{subarray}}{\varepsilon}_{j}\bigg{)}f_{\mathsf{ES}}\zeta_{\mathcal{A}}\binom{{\boldsymbol{\sl{s}}}}{{\boldsymbol{\varepsilon}}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\
2|j\end{subarray}}{\varepsilon}_{j}\bigg{)}\zeta_{\sharp}^{\mathcal{S}}\binom{{\boldsymbol{\sl{s}}}}{{\boldsymbol{\varepsilon}}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\
2|j\end{subarray}}{\varepsilon}_{j}\bigg{)}\sum_{i=0}^{d}\Big{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}{\varepsilon}_{\ell}\Big{)}\zeta_{\sharp}\binom{s_{i},\dots,s_{1}}{{\varepsilon}_{i},\dots,{\varepsilon}_{1}}\zeta_{\sharp}\binom{s_{i+1},\dots,s_{d}}{{\varepsilon}_{i+1},\dots,{\varepsilon}_{d}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{i=0}^{d}\bigg{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}\bigg{)}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{d}=\pm
1}\bigg{(}\prod_{\begin{subarray}{c}1\leq j\leq d\\\
2|j\end{subarray}}{\varepsilon}_{j}\bigg{)}\bigg{(}\prod_{\ell=1}^{i}{\varepsilon}_{\ell}\bigg{)}\zeta_{\sharp}\binom{s_{i},\dots,s_{1}}{{\varepsilon}_{i},\dots,{\varepsilon}_{1}}\zeta_{\sharp}\binom{s_{i+1},\dots,s_{d}}{{\varepsilon}_{i+1},\dots,{\varepsilon}_{d}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{i=0}^{d}\Big{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}\Big{)}\bigg{(}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{i}=\pm
1}\prod_{\begin{subarray}{c}1\leq j\leq i\\\ 2\nmid
j\end{subarray}}{\varepsilon}_{j}\zeta_{\sharp}\binom{s_{i},\dots,s_{1}}{{\varepsilon}_{i},\dots,{\varepsilon}_{1}}\bigg{)}$
$\displaystyle\,\hskip
170.71652pt\times\bigg{(}\sum_{{\varepsilon}_{1},\dots,{\varepsilon}_{i}=\pm
1}\prod_{\begin{subarray}{c}i<j\leq d\\\
2|j\end{subarray}}{\varepsilon}_{j}\zeta_{\sharp}\binom{s_{i+1},\dots,s_{d}}{{\varepsilon}_{i+1},\dots,{\varepsilon}_{d}}\bigg{)}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2^{d}}\sum_{i=0}^{d}\Big{(}\prod_{\ell=1}^{i}(-1)^{s_{\ell}}\Big{)}T_{\sharp}(s_{i},\dots,s_{1})T_{\sharp}(s_{i+1},\dots,s_{d})$
$\displaystyle=$
$\displaystyle\,T_{\sharp}^{\mathcal{S}}({\boldsymbol{\sl{s}}}).$

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
$\zeta(2)$, and vice versa. Sometimes, they are valid for the symmetric
version even without modulo $\zeta(2)$. For example, the following reversal
relations hold for both types of sums by [25, Propositions 2.8 and 2.9]. For
${\boldsymbol{\sl{s}}}=(s_{1},\dots,s_{d})$, we state
$\overleftarrow{{\boldsymbol{\sl{s}}}}=(s_{d},\dots,s_{1})$.

######  Proposition 2.2 (Reversal relations).

For all ${\boldsymbol{\sl{s}}}\in\mathbb{N}^{d}$, if $d$ is even, then

$\displaystyle T_{\mathcal{A}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}T_{\mathcal{A}}({\boldsymbol{\sl{s}}})\quad\text{and}\quad
S_{\mathcal{A}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=(-1)^{|{\boldsymbol{\sl{s}}}|}S_{\mathcal{A}}({\boldsymbol{\sl{s}}}),$
$\displaystyle T_{\ast}^{\mathcal{S}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}T_{\ast}^{\mathcal{S}}({\boldsymbol{\sl{s}}})\quad\text{and}\quad
S_{\ast}^{\mathcal{S}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=(-1)^{|{\boldsymbol{\sl{s}}}|}S_{\ast}^{\mathcal{S}}({\boldsymbol{\sl{s}}}),$

and if $d$ is odd, then

$\displaystyle T_{\mathcal{A}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}S_{\mathcal{A}}({\boldsymbol{\sl{s}}})\quad\text{and}\quad
S_{\mathcal{A}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=(-1)^{|{\boldsymbol{\sl{s}}}|}T_{\mathcal{A}}({\boldsymbol{\sl{s}}}),$
$\displaystyle T_{\ast}^{\mathcal{S}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}S_{\ast}^{\mathcal{S}}({\boldsymbol{\sl{s}}})\quad\text{and}\quad
S_{\ast}^{\mathcal{S}}(\overleftarrow{{\boldsymbol{\sl{s}}}})=(-1)^{|{\boldsymbol{\sl{s}}}|}T_{\ast}^{\mathcal{S}}({\boldsymbol{\sl{s}}}).$

##  3 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
$({\boldsymbol{\sl{s}}},{\boldsymbol{\sigma}})\in\mathbb{N}^{d}\times\\{\pm
1\\}^{d}$ (i.e., $(s_{1},{\sigma}_{1})\neq(1,1)$), we define the alternating
multiple $T$-values as

$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}}):=\sum_{\begin{subarray}{c}n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-d+j-1)/2}}{n_{j}^{s_{j}}}.$

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
$T^{\prime}({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}})$, then

$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}})=T^{\prime}({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}})\prod_{d-j\equiv
0,1{\ (\rm{mod}\ 2)}}{\sigma}_{j}.$ (7)

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 ${\sigma}_{j}=-1$. For example,

$T(\bar{2},1)=\sum_{n>m>0}\frac{(-1)^{n-1}}{(2n-2)^{2}(2m-1)}.$

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

$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}};x):=\sum_{\begin{subarray}{c}n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}x^{n_{1}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-d+j-1)/2}}{n_{j}^{s_{j}}}.$

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 $\eta_{1},\dots,\eta_{d}=\pm
1$, it is then easy to evaluate the iterated integral:

$\displaystyle\,\int_{0}^{x}\left(\frac{dt}{t}\right)^{s_{1}-1}\frac{dt}{1-\eta_{1}t^{2}}\cdots\left(\frac{dt}{t}\right)^{s_{d}-1}\frac{dt}{1-\eta_{d}t^{2}}$
$\displaystyle=$
$\displaystyle\,\sum_{k_{1}>\dots>k_{d}>0}x^{2(k_{1}+\dots+k_{d})+d}\prod_{j=1}^{d}\frac{\eta_{j}^{k_{j}}}{(2k_{j}+2k_{j+1}+\dots+2k_{d}+d-j+1)^{s_{j}}}.$

Let

${\texttt{y}}_{0}=\frac{dt}{t},\qquad{\texttt{y}}_{1}=\frac{dt}{1-t^{2}},\qquad{\texttt{y}}_{-1}:=\frac{dt}{1+t^{2}}.$

By changing the indices $n_{j}=2k_{j}+2k_{j+1}+\dots+2k_{d}+d-j+1$, we
immediately obtain

$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}};x)=\int_{0}^{x}{\bf
p}\Big{(}{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{{\sigma}_{1}}\cdots{\texttt{y}}_{0}^{s_{d}-1}{\texttt{y}}_{{\sigma}_{d}}\Big{)}:=\int_{0}^{x}{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{\eta_{1}}\cdots{\texttt{y}}_{0}^{s_{d}-1}{\texttt{y}}_{\eta_{d}},$

where $\eta_{j}={\sigma}_{1}\cdots{\sigma}_{j}$ for all $j\geq 1$ and ${\bf
p},{\bf q}$ represent the conversions between the series and the integral
expressions of alternating MTVs:

$\displaystyle{\bf p}({\bf u}):=$
$\displaystyle\,{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{{\sigma}_{1}}\dots\texttt{y}_{0}^{s_{j}-1}{\texttt{y}}_{{\sigma}_{1}\cdots{\sigma}_{j}}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{{\sigma}_{1}\cdots{\sigma}_{d}},$
(8) $\displaystyle{\bf q}({\bf u}):=$
$\displaystyle\,{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{{\sigma}_{1}}\dots\texttt{y}_{0}^{s_{j}-1}{\texttt{y}}_{{\sigma}_{j}/{\sigma}_{j-1}}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{{\sigma}_{d}/{\sigma}_{d-1}}.$
(9)

Namely, ${\bf p}$ pushes a word used in the series definition to a word used
in the integral expression, whereas ${\bf q}$ 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 ${\mathfrak{A}}_{1}^{*}$ (resp.
${\mathfrak{A}}_{2}^{*}$) be the $\mathbb{Q}$-algebra of words on
$\\{{\texttt{x}}_{0},{\texttt{x}}_{1}\\}$ (resp.
$\\{{\texttt{x}}_{0},{\texttt{x}}_{1},{\texttt{x}}_{-1}\\}$) with
concatenation as the product. Let ${\mathfrak{A}}_{j}^{1}$ ($j=1,2$) be the
sub-algebra generated by the words not ending with ${\texttt{x}}_{0}$. Then,
for each word ${\bf
u}={\texttt{x}}_{0}^{s_{1}-1}{\texttt{x}}_{\eta_{1}}\dots\texttt{x}_{0}^{s_{d}-1}{\texttt{x}}_{\eta_{d}}\in{\mathfrak{A}}_{2}^{1}$,
we define

$\zeta_{\mathcal{A}}({\bf
u}):=\zeta_{\mathcal{A}}(s_{1},\dots,s_{d};{\sigma}_{1},\dots,{\sigma}_{d})$

where ${\sigma}_{1}=\eta_{1}$ and ${\sigma}_{j}=\eta_{j}/\eta_{j-1}$ for all
$j\geq 2$. Set $\tau(1)=1$ and

$\tau({\texttt{x}}_{0}^{s_{1}-1}{\texttt{x}}_{1}\dots\texttt{x}_{0}^{s_{d}-1}{\texttt{x}}_{1})=(-1)^{s_{1}+\dots+s_{r}}{\texttt{x}}_{0}^{s_{d}-1}{\texttt{x}}_{1}\dots\texttt{x}_{0}^{s_{1}-1}{\texttt{x}}_{1}.$

######  Theorem 3.1 ([27, Theorem 8.4.3]).

For all words, ${\bf w},{\bf u}\in{\mathfrak{A}}_{1}^{1}$, ${\bf
v}\in{\mathfrak{A}}_{2}^{1}$, and $s\in\mathbb{N}$, we have

  * (i)

$\zeta_{\mathcal{A}}({\bf u}\shuffle{\bf v})=\zeta_{\mathcal{A}}(\tau({\bf
u}){\bf v})$;

  * (ii)

$\zeta_{\mathcal{A}}(({\bf w}{\bf u})\shuffle{\bf v})=\zeta_{\mathcal{A}}({\bf
u}\shuffle\tau({\bf w}){\bf v})$;

  * (iii)

$\zeta_{\mathcal{A}}(({\emph{{x}}}_{0}^{s-1}{\emph{{x}}}_{1}{\bf
u})\shuffle{\bf v})=(-1)^{s}\zeta_{\mathcal{A}}({\bf
u}\shuffle({\emph{{x}}}_{0}^{s-1}{\emph{{x}}}_{1}{\bf v})).$

For alternating MTVs, we can similarly let ${\mathfrak{T}}_{1}^{*}$ (resp.
${\mathfrak{T}}_{2}^{*}$) be the $\mathbb{Q}$-algebra of words on
$\\{{\texttt{y}}_{0},{\texttt{y}}_{1}\\}$ (resp.
$\\{{\texttt{y}}_{0},{\texttt{y}}_{1},{\texttt{y}}_{-1}\\}$) with
concatenation as the product. Let ${\mathfrak{T}}_{j}^{1}$ ($j=1,2$) be the
sub-algebra generated by the words not ending with ${\texttt{y}}_{0}$. Then,
for each word ${\bf
u}={\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{{\sigma}_{1}}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{{\sigma}_{d}}\in{\mathfrak{A}}_{2}^{1}$,
let ${\bf p},{\bf q}:{\mathfrak{A}}_{2}^{1}\to{\mathfrak{A}}_{2}^{1}$ 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:

$F_{\ast}({\bf u}):=F({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}}),\quad
F_{\shuffle}({\bf u}):=F_{\ast}\big{(}{\bf q}({\bf u})\big{)},\quad
F_{\ast}({\bf u})=F_{\shuffle}\big{(}{\bf p}({\bf u})\big{)},$

where $F(-)$ can be either $T(-)$, $T_{\mathcal{A}}$, or $T(-;x)$ or even
their partial sums, such as

$T_{n}({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}}):=\sum_{\begin{subarray}{c}n>n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-d+j-1)/2}}{n_{j}^{s_{j}}}.$

For all words ${\bf w}\in{\mathfrak{T}}_{2}^{1}$, we set $T_{\mathcal{A}}({\bf
w}):=T_{\mathcal{A},\shuffle}({\bf w})=T_{\mathcal{A},\ast}\big{(}{\bf q}({\bf
w})\big{)}.$ Further, set $\tau(1)=1$ and

$\tau({\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{1}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{1})=(-1)^{s_{1}+\dots+s_{r}}{\texttt{y}}_{0}^{s_{d}-1}{\texttt{y}}_{1}\dots\texttt{y}_{0}^{s_{1}-1}{\texttt{y}}_{1}.$

######  Theorem 3.2.

For all words ${\bf w},{\bf u}\in{\mathfrak{T}}_{1}^{1}$, ${\bf
v}\in{\mathfrak{T}}_{2}^{1}$ and $s\in\mathbb{N}$, we have

  * (i)

$T_{\mathcal{A}}({\bf u}\shuffle{\bf v})=T_{\mathcal{A}}(\tau({\bf u}){\bf
v})$ if $\operatorname*{dep}({\bf u})+\operatorname*{dep}({\bf v})$ is even;

  * (ii)

$T_{\mathcal{A}}(({\bf w}{\bf u})\shuffle{\bf v})=T_{\mathcal{A}}({\bf
u}\shuffle\tau({\bf w}){\bf v})$ if $\operatorname*{dep}({\bf
u})+\operatorname*{dep}({\bf v})+\operatorname*{dep}({\bf w})$ is even;

  * (iii)

$T_{\mathcal{A}}(({\emph{{y}}}_{0}^{s-1}{\emph{{y}}}_{1}{\bf u})\shuffle{\bf
v})=(-1)^{s}T_{\mathcal{A}}({\bf
u}\shuffle({\emph{{y}}}_{0}^{s-1}{\emph{{y}}}_{1}{\bf v}))$ if
$\operatorname*{dep}({\bf u})+\operatorname*{dep}({\bf v})$ is odd.

###### Proof.

By taking ${\bf u}=\emptyset$ and then setting ${\bf w}={\bf u}$, we see that
(ii) implies (i). By decomposing ${\bf w}$ into strings of type
${\texttt{y}}_{0}^{s-1}{\texttt{y}}_{1}$, we see that (iii) implies (ii). So,
we only need to prove (iii).

For simplicity, write ${\texttt{a}}={\texttt{y}}_{0}$ and
${\texttt{b}}={\texttt{y}}_{1}$ for the rest of this proof. Observe that for
any odd prime $p$, the coefficient of $x^{p}$ of
$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}};x)$ is nontrivial if and only
if $\operatorname*{dep}({\boldsymbol{\sl{s}}})$ is odd. Therefore, if the
depth $d$ of the word ${\bf w}$ is even, the coefficient of $x^{p}$ in
$T_{\ast}\big{(}{\bf q}({\texttt{b}}{\bf w});x\big{)}$ is given as

${\rm Coeff}_{x^{p}}\Big{[}T_{\ast}\big{(}{\bf q}({\texttt{b}}{\bf
w});x\big{)}\Big{]}={\rm
Coeff}_{x^{p}}\Big{[}T_{\shuffle}\big{(}{\texttt{b}}{\bf
w};x\big{)}\Big{]}=\frac{1}{p}T_{p,\shuffle}({\bf w})$

since ${\bf q}({\texttt{b}}{\bf w})={\texttt{b}}{\bf q}({\bf w})$. Observe
that

${\texttt{b}}\Big{(}({\texttt{a}}^{s-1}{\texttt{b}}{\bf u})\shuffle{\bf
v}-(-1)^{s}{\bf u}\shuffle({\texttt{a}}^{s-1}{\texttt{b}}{\bf
v})\Big{)}=\sum_{i=0}^{s-1}(-1)^{i}({\texttt{a}}^{s-1-i}{\texttt{b}}{\bf
u})\shuffle({\texttt{a}}^{i}{\texttt{b}}{\bf v}).$

Hence, if $\operatorname*{dep}({\bf u})+\operatorname*{dep}({\bf v})$ is odd,
then, by first applying $T_{\shuffle}(-;x)$ to the above and then extracting
the coefficients of $x^{p}$ from both sides, we obtain

$\displaystyle\,\frac{1}{p}\Big{(}T_{p,\shuffle}\big{(}({\texttt{a}}^{s-1}{\texttt{b}}{\bf
u})\shuffle{\bf v}\big{)}-(-1)^{s}T_{p,\shuffle}\big{(}{\bf
u}\shuffle({\texttt{a}}^{s-1}{\texttt{b}}{\bf v})\big{)}\Big{)}$
$\displaystyle=$ $\displaystyle\,\sum_{i=0}^{s-1}(-1)^{i}{\rm
Coeff}_{x^{p}}\big{[}T_{\shuffle}({\texttt{a}}^{s-1-i}{\texttt{b}}{\bf
u};x)T_{\shuffle}({\texttt{a}}^{i}{\texttt{b}}{\bf v};x)\big{]}$
$\displaystyle=$ $\displaystyle\,\sum_{i=0}^{s-1}(-1)^{i}\sum_{j=1}^{p-1}{\rm
Coeff}_{x^{j}}\big{[}T_{\shuffle}({\texttt{a}}^{s-1-i}{\texttt{b}}{\bf
u};t)\big{]}\cdot{\rm
Coeff}_{x^{p-j}}\big{[}T_{\shuffle}({\texttt{a}}^{i}{\texttt{b}}{\bf
v};t)\big{]}$

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

$T_{p,\shuffle}({\texttt{a}}^{s-1}{\texttt{b}}{\bf u})\shuffle{\bf
v})\equiv(-1)^{s}T_{p,\shuffle}({\bf
u}\shuffle({\texttt{a}}^{s-1}{\texttt{b}}{\bf v}))\pmod{p}$

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 $\Psi_{2}$ and
considering the words of the form
${\texttt{x}}_{0}^{s_{1}-1}({\texttt{x}}_{1}+{\texttt{x}}_{-1})\cdots{\texttt{x}}_{0}^{s_{d}-1}({\texttt{x}}_{1}+{\texttt{x}}_{-1})$
in [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 $d\in\mathbb{N}$ is odd. For all $s_{1},\dots,s_{d}\in\mathbb{N}$, we
have

$T_{\mathcal{A}}(1,{\boldsymbol{\sl{s}}})+T_{\mathcal{A}}({\boldsymbol{\sl{s}}},1)+\sum_{j=1}^{d}\sum_{a=1}^{s_{j}+1}T_{\mathcal{A}}(s_{1},\dots,s_{j-1},a,s_{j}+1-a,s_{j+1},\dots,s_{d})=0.$

###### Proof.

This follows immediately from the linear shuffle relation

$T_{\mathcal{A}}({\texttt{y}}_{1}\shuffle{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{1}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{1})=-T_{\mathcal{A}}({\texttt{y}}_{1}{\texttt{y}}_{0}^{s_{1}-1}{\texttt{y}}_{1}\dots\texttt{y}_{0}^{s_{d}-1}{\texttt{y}}_{1})$

by taking $s_{0}=1$ and ${\bf u}=1$ in Theorem 3.2(iii). ∎

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

######  Conjecture 3.5.

For all $k\in\mathbb{N}$, we have

$T_{\mathcal{A}}(2,\\{1\\}^{k})=\frac{(-1)^{k}}{2^{k-1}}T_{\mathcal{A}}(1,k+1),\quad
T_{\shuffle}^{\mathcal{S}}(\\{1\\}^{w})=\frac{(-1)^{k}}{2^{k-1}}T_{\shuffle}^{\mathcal{S}}(1,k+1).$

######  Proposition 3.6.

If $k$ is odd, then for all $\ell\leq k$, we have

$T_{\mathcal{A}}(\\{1\\}^{\ell},2,\\{1\\}^{k-\ell})=\frac{(-1)^{\ell}}{\ell+1}\binom{k+1}{\ell}T_{\mathcal{A}}(2,\\{1\\}^{k}).$
(10)

If, in addition, we assume Conjecture 3.5 holds, then

$T_{\mathcal{A}}(\\{1\\}^{\ell},2,\\{1\\}^{k-\ell})=\frac{(-1)^{\ell+k}}{2^{k-1}(\ell+1)}\binom{k+1}{\ell}T_{\mathcal{A}}(1,k+1).$
(11)

###### Proof.

For all $\ell\leq k$, we see the linear shuffle relations

$T_{\mathcal{A}}({\texttt{y}}_{1}\shuffle{\texttt{y}}_{1}^{\ell}{\texttt{y}}_{0}{\texttt{y}}_{1}^{k-\ell})=-{\texttt{y}}_{1}^{\ell+1}{\texttt{y}}_{0}{\texttt{y}}_{1}^{k-\ell}.$

Thus, by setting
$a_{\ell}=T_{\mathcal{A}}(\\{1\\}^{\ell},2,\\{1\\}^{k-\ell})$, we obtain

$(\ell+2)a_{\ell+1}+(k-\ell+1)a_{\ell}=0.$

Hence,

$\displaystyle a_{\ell+1}=$
$\displaystyle\,-\frac{k-\ell+1}{\ell+2}a_{\ell}=\frac{(k-\ell+1)(k-\ell+2)}{(\ell+2)(\ell+1)}a_{\ell-1}=\cdots$
$\displaystyle=$
$\displaystyle\,(-1)^{\ell-1}\frac{(k-\ell+1)(k-\ell+2)\cdots(k+1)}{(\ell+2)(\ell+1)\cdots
2}a_{0}$ $\displaystyle=$
$\displaystyle\,(-1)^{\ell-1}\frac{(k+1)!}{(\ell+2)!(k-\ell)!}a_{0}=\frac{(-1)^{\ell-1}}{\ell+2}\binom{k+1}{\ell+1}a_{0},$

which yields (10). Then, (11) follows immediately if we assume Conjecture 3.5.
∎

###  3.1 Values at Small Depths/Weights

First, we observe that since $\zeta_{\mathcal{A}}(s)=0$ for all
$s\in\mathbb{N}$ according to [27, Theorem 8.2.7] we must have

$\displaystyle
S_{\mathcal{A}}(s)=-T_{\mathcal{A}}(s)=\frac{1}{2}\zeta_{\mathcal{A}}(\bar{s})=$
$\displaystyle\,\left\\{\begin{array}[]{ll}-{\texttt{q}}_{2},&\quad\hbox{if
$s=1$;}\\\ (2^{1-s}-1)\beta_{s},&\quad\hbox{if $s\geq 2$,}\end{array}\right.$
(14)

where ${\texttt{q}}_{2}$ is the Fermat quotient (6), and $\beta_{s}$ is given
in (5). Further, in depth two, according to Proposition 2.6 in our arxiv paper
2402.08160, we see that for all $a,b\in\mathbb{N}$, if $w=a+b$ is odd, then

$\displaystyle S_{\mathcal{A}}(a,b)=$
$\displaystyle\,T_{\mathcal{A}}(a,b)=\frac{(-1)^{a}}{2}\Big{(}1-2^{-w}\Big{)}\binom{w}{a}\beta_{w}.$
(15)

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

$\displaystyle\zeta_{\mathcal{A}}(1,1,1)=0,\quad\zeta_{\mathcal{A}}(\bar{1},\bar{1},\bar{1})=$
$\displaystyle\,-\frac{4}{3}{\emph{{q}}}_{p}^{3}-\frac{\beta_{3}}{2},\quad\zeta_{\mathcal{A}}(1,1,\bar{1})=\zeta_{\mathcal{A}}(\bar{1},1,1)=-\frac{{\emph{{q}}}^{3}}{3}-\frac{7}{8}\beta_{3},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},1,\bar{1})=0,\quad\zeta_{\mathcal{A}}(1,\bar{1},1)=$
$\displaystyle\,\frac{2{\emph{{q}}}^{3}}{3}+\frac{\beta_{3}}{4},\quad\zeta_{\mathcal{A}}(\bar{1},\bar{1},1)=-\zeta_{\mathcal{A}}(1,\bar{1},\bar{1})=-{\emph{{q}}}_{p}^{3}-\frac{21}{8}\beta_{3}.$

###### Proof.

It immediately follows on from [19, Propositions 7.3 and 7.6] that

$\displaystyle\zeta_{\mathcal{A}}(\bar{1},\bar{1},\bar{1})=$
$\displaystyle\,-\frac{4}{3}{\texttt{q}}_{p}^{3}-\frac{1}{2}\beta_{3},\quad\zeta_{\mathcal{A}}(1,\bar{1},1)=-2\zeta_{\mathcal{A}}(\bar{1},1,1)-\frac{3}{2}\beta_{3},\quad\zeta_{\mathcal{A}}(\bar{1},1,\bar{1})=0,$
$\displaystyle\zeta_{\mathcal{A}}(1,1,\bar{1})=$
$\displaystyle\,\zeta_{\mathcal{A}}(\bar{1},1,1),\quad\zeta_{\mathcal{A}}(\bar{1},\bar{1},1)=-\zeta_{\mathcal{A}}(1,\bar{1},\bar{1})=-{\texttt{q}}_{p}^{3}-\frac{21}{8}\beta_{3}.$

According to the linear shuffle relations for finite Euler sums, we have

$\displaystyle-\zeta_{\mathcal{A}}({\texttt{b}}{\texttt{c}}{\texttt{c}})=\zeta_{\mathcal{A}}({\texttt{b}}\shuffle{\texttt{c}}{\texttt{c}})=$
$\displaystyle\zeta_{\mathcal{A}}({\texttt{b}}{\texttt{c}}{\texttt{c}})+\zeta_{\mathcal{A}}({\texttt{c}}{\texttt{b}}{\texttt{c}})+\zeta_{\mathcal{A}}({\texttt{c}}{\texttt{c}}{\texttt{b}})$

which readily yields the identity

$2\zeta_{\mathcal{A}}(1,\bar{1},1)+\zeta_{\mathcal{A}}(\bar{1},\bar{1},\bar{1})+\zeta_{\mathcal{A}}(\bar{1},1,\bar{1})=0,$
(16)

which in turn quickly implies all the evaluations in the proposition. ∎

######  Corollary 3.8.

We have

$T_{\mathcal{A}}(1,1,1)=-S_{\mathcal{A}}(1,1,1)=\frac{3}{16}\beta_{3}.$

###### Proof.

The corollary is an immediate consequence of the definitions using Proposition
3.7. Alternatively, we can prove it directly, as follows: since
$\zeta_{\mathcal{A}}(1,\bar{1},\bar{1})=-\zeta_{\mathcal{A}}(\bar{1},\bar{1},1)$
from a reversal, and $\zeta_{\mathcal{A}}(1,1,1)=0$, we obtain

$\displaystyle 8T_{\mathcal{A}}(1,1,1)=$
$\displaystyle\,\zeta_{\mathcal{A}}(\bar{1},1,\bar{1})+\zeta_{\mathcal{A}}(\bar{1},\bar{1},\bar{1})+\zeta_{\mathcal{A}}(1,\bar{1},1)-$
$\displaystyle\zeta_{\mathcal{A}}(1,1,\bar{1})-\zeta_{\mathcal{A}}(\bar{1},1,1)$
$\displaystyle=$
$\displaystyle\,-\zeta_{\mathcal{A}}(1,\bar{1},1)-2\zeta_{\mathcal{A}}(1,1,\bar{1})\quad$
$\displaystyle(\text{by \eqref{equ:linShuESbcc}})$ $\displaystyle=$
$\displaystyle\,\zeta_{\mathcal{A}}(\bar{2},1)+\zeta_{\mathcal{A}}(\bar{1},2)-\zeta_{\mathcal{A}}(1)\zeta_{\mathcal{A}}(1,\bar{1})\quad$
$\displaystyle(\text{by shuffle})$ $\displaystyle=$
$\displaystyle\,\frac{3}{2}\beta_{3}.$

according to [27, Theorem 8.6.4]. ∎

######  Proposition 3.9.

We have

$T_{\shuffle}^{\mathcal{S}}(1,1,1)=-S_{\shuffle}^{\mathcal{S}}(1,1,1)=\frac{3}{16}\zeta(3).$

###### Proof.

The weighted three Euler sums are all expressible in terms of
$\zeta(\bar{2},1),$ $\zeta(\bar{1},1,1)$ and $\zeta(\bar{1},2)$ by [27,
Proposition 14.2.7]. Hence, one easily deduces that

$\displaystyle\zeta_{\shuffle}^{\mathcal{S}}(1,1,1)=$
$\displaystyle\,\zeta_{\shuffle}^{\mathcal{S}}(\bar{1},1,\bar{1})=0,$
$\displaystyle\zeta_{\shuffle}^{\mathcal{S}}(\bar{1},1,1)=$
$\displaystyle\,\zeta_{\shuffle}^{\mathcal{S}}(1,1,\bar{1})=\zeta(\bar{1},1,1)+\zeta(\bar{1})\zeta_{\shuffle}(1,1)-\zeta_{\shuffle}(1,\bar{1})\zeta_{\shuffle}(1)+\zeta_{\shuffle}(1,1,\bar{1})$
$\displaystyle=$
$\displaystyle\,\zeta(\bar{1},1,1)+\zeta(\bar{1})\frac{T^{2}}{2}-\Big{(}\zeta(\bar{1})T-\zeta(\bar{1},\bar{1})\Big{)}T+\zeta(\bar{1})\frac{T^{2}}{2}-\zeta(\bar{1},\bar{1})T+\zeta(\bar{1},\bar{1},1)$
$\displaystyle=$ $\displaystyle\,\zeta(\bar{1},1,1)+\zeta(\bar{1},\bar{1},1),$
$\displaystyle\zeta_{\shuffle}^{\mathcal{S}}(\bar{1},\bar{1},1)=$
$\displaystyle\,-\zeta_{\shuffle}^{\mathcal{S}}(1,\bar{1},\bar{1})=3\zeta(\bar{1},\bar{1},1)+3\zeta(\bar{1},1,1),$
$\displaystyle\zeta_{\shuffle}^{\mathcal{S}}(1,\bar{1},1)=$
$\displaystyle\,2\zeta_{\shuffle}(1,\bar{1},1)-2\zeta(\bar{1},1)\zeta_{\shuffle}(1)=-2\zeta(\bar{1},\bar{1},\bar{1})-2\zeta(\bar{1},1,\bar{1}),$
$\displaystyle\zeta_{\shuffle}^{\mathcal{S}}(\bar{1},\bar{1},\bar{1})=$
$\displaystyle\,2\zeta(\bar{1},\bar{1},\bar{1})+2\zeta(\bar{1})\zeta(\bar{1},\bar{1})=4\zeta(\bar{1},\bar{1},\bar{1})+4\zeta(\bar{1},1,\bar{1}).$

In [27, Proposition 14.2.7], we have

$\displaystyle\zeta(3)=$
$\displaystyle\,8\zeta(\bar{2},1),\quad\zeta(\bar{1},\bar{1},1)=\zeta(\bar{1},2)-5\zeta(\bar{2},1)+\zeta(\bar{1},1,1),$
$\displaystyle\zeta(\bar{1},1,\bar{1})=$
$\displaystyle\,\zeta(\bar{2},1)+\zeta(\bar{1},1,1),\quad\zeta(\bar{1},\bar{1},\bar{1})=\zeta(\bar{1},2)+\zeta(\bar{1},1,1).$

Thus, we get

$\displaystyle T_{\shuffle}^{\mathcal{S}}(1,1,1)=$
$\displaystyle\,\frac{1}{4}\Big{(}\zeta(\bar{1},\bar{1},\bar{1})+\zeta(\bar{1},1,\bar{1})-\zeta(\bar{1},1,1)-\zeta(\bar{1},\bar{1},1)\Big{)}=\frac{6}{4}\zeta(\bar{2},1)=\frac{3}{16}\zeta(3),$
$\displaystyle S_{\shuffle}^{\mathcal{S}}(1,1,1)=$
$\displaystyle\,-\frac{1}{4}\Big{(}\zeta(\bar{1},\bar{1},\bar{1})+\zeta(\bar{1},1,\bar{1})-\zeta(\bar{1},1,1)-\zeta(\bar{1},\bar{1},1)\Big{)}=-\frac{3}{16}\zeta(3),$

as desired. ∎

In general, we can use linear shuffles to derive many relations from the
finite Euler sums. For example,

$\displaystyle{\texttt{b}}\shuffle{\texttt{a}}{\texttt{c}}{\texttt{b}}:$
$\displaystyle\,2\zeta_{\mathcal{A}}(1,\bar{2},\bar{1})+\zeta_{\mathcal{A}}(2,\bar{1},\bar{1})+2\zeta_{\mathcal{A}}(\bar{2},\bar{1},1)=0,$
$\displaystyle{\texttt{b}}\shuffle{\texttt{a}}{\texttt{c}}{\texttt{c}}:$
$\displaystyle\,2\zeta_{\mathcal{A}}(1,\bar{2},1)+\zeta_{\mathcal{A}}(2,\bar{1},1)+\zeta_{\mathcal{A}}(\bar{2},\bar{1},\bar{1})+\zeta_{\mathcal{A}}(\bar{2},1,\bar{1})=0,$
$\displaystyle{\texttt{a}}{\texttt{b}}\shuffle{\texttt{b}}{\texttt{c}}:$
$\displaystyle\,3\zeta_{\mathcal{A}}(2,1,\bar{1})+\zeta_{\mathcal{A}}(2,\bar{1},\bar{1})+\zeta_{\mathcal{A}}(1,\bar{2},\bar{1})+\zeta_{\mathcal{A}}(1,2,\bar{1})+\zeta_{\mathcal{A}}(1,\bar{1},\bar{2})=0,$
$\displaystyle{\texttt{a}}{\texttt{b}}\shuffle{\texttt{c}}{\texttt{b}}:$
$\displaystyle\,2\zeta_{\mathcal{A}}(2,\bar{1},\bar{1})+2\zeta_{\mathcal{A}}(\bar{2},\bar{1},1)+2\zeta_{\mathcal{A}}(\bar{1},\bar{2},1)+\zeta_{\mathcal{A}}(\bar{1},\bar{1},2)=0,$
$\displaystyle{\texttt{a}}{\texttt{b}}\shuffle{\texttt{c}}{\texttt{c}}:$
$\displaystyle\,2\zeta_{\mathcal{A}}(2,\bar{1},1)+\zeta_{\mathcal{A}}(\bar{2},\bar{1},\bar{1})+\zeta_{\mathcal{A}}(\bar{2},1,\bar{1})+\zeta_{\mathcal{A}}(\bar{1},\bar{2},\bar{1})+\zeta_{\mathcal{A}}(\bar{1},2,\bar{1})+\zeta_{\mathcal{A}}(\bar{1},1,\bar{2})=0,$
$\displaystyle{\texttt{b}}\shuffle{\texttt{b}}{\texttt{a}}{\texttt{c}}^{2}:$
$\displaystyle\,3\zeta_{\mathcal{A}}(1,1,\bar{2},1)+\zeta_{\mathcal{A}}(1,2,\bar{1},1)+\zeta_{\mathcal{A}}(1,\bar{2},\bar{1},\bar{1})+\zeta_{\mathcal{A}}(1,\bar{2},1,\bar{1})=0,$
$\displaystyle{\texttt{b}}\shuffle{\texttt{c}}^{4}:$
$\displaystyle\,2\zeta_{\mathcal{A}}(1,\bar{1},1^{3})+\zeta_{\mathcal{A}}(\bar{1}^{3},1,1)+\zeta_{\mathcal{A}}(\bar{1},1,\bar{1}^{2},1)+\zeta_{\mathcal{A}}(\bar{1},1^{2},\bar{1}^{2})+\zeta_{\mathcal{A}}(\bar{1},1^{3},\bar{1})=0.$

We can also use reversal and shuffle relations to express all finite Euler
sums of weight up to 6 according to the explicitly given basis in each weight.
Aided by Maple computation, we arrive at the following main theorem on the
structure of finite Euler sums of a lower weight.

######  Theorem 3.10.

Let $\mathsf{FES}_{w}$ be the $\mathbb{Q}$-vector space generated by finite
Euler sums of weight $w$. Then, we have the following generating sets for
$w<7$:

$\displaystyle\mathsf{FES}_{1}=$
$\displaystyle\,\langle{\emph{{q}}}_{2}\rangle,\quad\mathsf{FES}_{2}=\langle{\emph{{q}}}_{2}^{2}\rangle,\quad\mathsf{FES}_{3}=\langle{\emph{{q}}}_{2}^{3},\beta_{3}\rangle,\quad\mathsf{FES}_{4}=\langle{\emph{{q}}}_{2}^{4},{\emph{{q}}}_{2}\beta_{3},\zeta_{\mathcal{A}}(1,\bar{3})\rangle,$
$\displaystyle\mathsf{FES}_{5}=$
$\displaystyle\,\langle{\emph{{q}}}_{2}^{5},{\emph{{q}}}_{2}^{2}\beta_{3},\beta_{5},\zeta_{\mathcal{A}}(\bar{1},2,2),\zeta_{\mathcal{A}}(\bar{1},\bar{2},2)\rangle,$
$\displaystyle\mathsf{FES}_{6}=$
$\displaystyle\,\langle{\emph{{q}}}_{2}^{6},{\emph{{q}}}_{2}^{3}\beta_{3},\beta_{3}^{2},{\emph{{q}}}_{2}\beta_{5},\zeta_{\mathcal{A}}(\bar{1},1,2,2),\zeta_{\mathcal{A}}(\bar{1},2,2,1),\zeta_{\mathcal{A}}(\bar{1},2,1,2),\zeta_{\mathcal{A}}(\bar{1},\\{1\\}^{3},2)\rangle.$

Let $\\{F_{k}\\}_{k\geq 0}$ be the Fibonnacci sequence defined by
$F_{0}=F_{1}=1$ and $F_{k}=F_{k-1}+F_{k-2}$ for all $k\geq 2$. Then, Theorem
3.10 provides strong support for the next conjecture.

######  Conjecture 3.11.

For every positive integer $w$, the $\mathbb{Q}$-space $\mathsf{FES}_{w}$ has
the following basis:

$\Big{\\{}\zeta_{\mathcal{A}}(\bar{1},b_{2},\dots,b_{d}):d\geq 0,b_{j}=1\text{
or }2,1+b_{2}+\dots+b_{d}=w\Big{\\}}.$

Consequently, $\dim_{\mathbb{Q}}\mathsf{FES}_{w}=F_{w-1}$ for all $w\geq 1$.

One may compare this to the conjecture on the ordinary Euler sums proposed by
Zlobin [27, Conjecture 14.2.3].

######  Conjecture 3.12.

For every positive integer $w$ the $\mathbb{Q}$-space $\mathsf{ES}_{w}$ has
the following basis:

$\Big{\\{}\zeta(\overline{b_{1}},b_{2},\dots,b_{d}):d\geq 1,b_{j}=1\text{ or
}2,b_{1}+b_{2}+\dots+b_{d}=w\Big{\\}}.$

Consequently, $\dim_{\mathbb{Q}}\mathsf{ES}_{w}=F_{w}$ for all $w\geq 1$.

Theorem 3.10 implies that the set in Conjecture 3.11 is a generating set for
all $w<7$ since

$\displaystyle\zeta_{\mathcal{A}}(\bar{1},1)=$
$\displaystyle\,-2{\texttt{q}}_{2},\quad\zeta_{\mathcal{A}}(\bar{1},1)={\texttt{q}}_{2}^{2},\quad\zeta_{\mathcal{A}}(\bar{1},2)=\tfrac{3}{4}\beta_{3},\quad\zeta([1,\bar{1},1)=\tfrac{2}{4}{\texttt{q}}_{2}^{3}+\tfrac{1}{4}\beta_{3},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},1,2)=$
$\displaystyle\,\tfrac{9}{4}{\texttt{q}}_{2}\beta_{3}-\zeta_{\mathcal{A}}(1,\bar{3}),\quad\zeta_{\mathcal{A}}(\bar{1},\\{1\\}^{3})=\tfrac{1}{12}{\texttt{q}}_{2}^{4}+\tfrac{7}{8}{\texttt{q}}_{2}\beta_{3}+\tfrac{1}{4}\zeta_{\mathcal{A}}(1,\bar{3}),$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},2,1)=$
$\displaystyle\,\tfrac{1}{2}\zeta_{\mathcal{A}}(1,\bar{3})-\tfrac{12}{4}{\texttt{q}}_{2}\beta_{3},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},2,1,1)=$
$\displaystyle\,\tfrac{695}{128}\beta_{5}-\tfrac{5}{4}\zeta_{\mathcal{A}}(\bar{1},2,2)-2\zeta_{\mathcal{A}}(\bar{1},1,1,2)-\tfrac{9}{4}{\texttt{q}}_{2}^{2}\beta_{3},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{\\{}1\\}^{4},1)=$
$\displaystyle\,-\tfrac{1}{60}{\texttt{q}}_{2}^{5}-\tfrac{23}{24}{\texttt{q}}_{2}^{2}\beta_{3}-\tfrac{1}{8}\zeta_{\mathcal{A}}(\bar{1},2,2)-\tfrac{1}{2}\zeta_{\mathcal{A}}(\bar{1},1,1,2)-\tfrac{25}{256}\beta_{5},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},1,2,1)=$
$\displaystyle\,\tfrac{33}{8}{\texttt{q}}_{2}^{2}\beta_{3}-\tfrac{555}{128}\beta_{5}+\tfrac{5}{4}\zeta_{\mathcal{A}}(\bar{1},2,2)+2\zeta_{\mathcal{A}}(\bar{1},1,1,2),$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},1,2,1,1)=$
$\displaystyle\,-\tfrac{1}{2}A+2B+C+D+\tfrac{9}{4}\beta_{3}^{2}+\tfrac{5}{8}{\texttt{q}}_{2}^{3}\beta_{3}+\tfrac{205}{64}{\texttt{q}}_{2}\beta_{5},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{\\{}1\\}^{4},2)=$
$\displaystyle\,-\tfrac{3}{4}A+\tfrac{19}{8}B+\tfrac{1}{4}C+D+\tfrac{201}{32}\beta_{3}^{2}+{\texttt{q}}_{2}^{3}\beta_{3}-\tfrac{645}{256}{\texttt{q}}_{2}\beta_{5},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},2,\\{1\\}^{3})=$
$\displaystyle\,\tfrac{1}{2}A-\tfrac{19}{8}B-\tfrac{5}{4}C-2D-\tfrac{1113}{256}\beta_{3}^{2}-\tfrac{5}{4}{\texttt{q}}_{2}^{3}\beta_{3}-\tfrac{1685}{256}{\texttt{q}}_{2}\beta_{5},$
$\displaystyle\zeta_{\mathcal{A}}(\bar{1},\\{1\\}^{5})=$
$\displaystyle\,\tfrac{1}{4}A-\tfrac{13}{16}B-\tfrac{1}{8}C-\tfrac{1}{2}D-\tfrac{1}{6}{\texttt{q}}_{2}^{3}\beta_{3}+\tfrac{817}{512}{\texttt{q}}_{2}\beta_{5}-\tfrac{811}{512}\beta_{3}^{2}+\tfrac{1}{360}{\texttt{q}}_{2}^{6},$

where
$A=\zeta_{\mathcal{A}}(\bar{1},1,2,2),B=\zeta_{\mathcal{A}}(\bar{1},2,1,2),C=\zeta_{\mathcal{A}}(\bar{1},2,1,2),$
and $D=\zeta_{\mathcal{A}}(\bar{1},2,2,1)$.

By using the evaluations of finite Euler sums, we can find all FMTVs of weight
less than 7. For example, we have

$\displaystyle T_{\mathcal{A}}(1,1,2)=$
$\displaystyle\,-\frac{1}{8}\zeta_{\mathcal{A}}(1,\bar{3})-\frac{21}{16}{\texttt{q}}_{2}\beta_{3},$
$\displaystyle T_{\mathcal{A}}(1,2,2)=$
$\displaystyle\,-\frac{1605}{256}\beta_{5}+\tfrac{9}{2}{\texttt{q}}_{2}^{2}\beta_{3}+3\zeta_{\mathcal{A}}(\bar{1},1,1,2).$

We then have the following structural theorem for these FMTVs:

######  Theorem 3.13.

Let $\mathsf{FMT}_{w}$ be the $\mathbb{Q}$-vector space generated by FMTVs of
weight $w$. Then, we have the following generating sets for $w<7$:

$\displaystyle\mathsf{FMT}_{1}=$
$\displaystyle\,\langle{\emph{{q}}}_{2}\rangle,\quad\mathsf{FMT}_{2}=\langle
0\rangle,\quad\mathsf{FMT}_{3}=\langle\beta_{3}\rangle,\quad\mathsf{FMT}_{4}=\langle{\emph{{q}}}_{2}\beta_{3},\zeta_{\mathcal{A}}(1,\bar{3})\rangle,$
$\displaystyle\mathsf{FMT}_{5}=$
$\displaystyle\,\langle\beta_{5},\zeta_{\mathcal{A}}(\bar{1},2,2),\zeta_{\mathcal{A}}(\bar{1},1,1,2)\rangle,\quad\mathsf{FMT}_{6}=\langle\beta_{3}^{2},{\emph{{q}}}_{2}\beta_{5},\zeta_{\mathcal{A}}(\bar{1},2,1,2)\rangle.$

Moreover, by using numerical computation aided by Maple (see [27, Appendix D],
for the pseudo codes), we can find a generating set of $\mathsf{FMT}_{w}$ for
every $w\leq 13.$ We will list the corresponding dimensions at the end of this
paper.

###  3.2 Homogeneous Cases

In this subsection, we will compute finite Euler sums
$\zeta({\boldsymbol{\sl{s}}})$ when ${\boldsymbol{\sl{s}}}$ is homogeneous,
i.e., ${\boldsymbol{\sl{s}}}=(\\{s\\}^{d})$ for some $s\in\mathbb{D}$. Then,
we will consider the corresponding results for FMTVs.

######  Proposition 3.14.

Let $\mathbb{N}_{\text{odd}}$ be the set of odd positive integers. For any
$d,s\in\mathbb{N}$, we have

$\zeta_{\mathcal{A}}(\\{\bar{s}\\}^{d})\in\sum_{\begin{subarray}{c}k_{0}\in\mathbb{N},\,k_{1},\dots,k_{\ell}\in\mathbb{N}_{\text{odd}}\\\
\delta_{s,1}k_{0}+k_{1}+\dots+k_{\ell}=d\end{subarray}}{\emph{{q}}}_{2}^{\delta_{s,1}k_{0}}\beta_{sk_{j}}\cdots\beta_{sk_{j}}\mathbb{Q},$

where $\delta_{s,1}$ is the Kronecker symbol. In particular,
$\zeta_{\mathcal{A}}(\\{\bar{s}\\}^{d})=0$ for all even $s$.

###### Proof.

Let $\Pi=(P_{1},\dots,P_{\ell})\in[d]$ denote any partition of $(1,\dots,d)$
into odd parts, i.e., all of $|P_{j}|$’s are odd numbers, where $|P_{j}|$ is
the cardinality of the set $P_{j}$. Let

$\mathcal{C}(\Pi)=(-1)^{d-\ell}(|P_{1}|-1)!\cdots(|P_{\ell}|-1)!.$

Observe that $\zeta_{\mathcal{A}}(\bar{n})=\zeta_{\mathcal{A}}(n)=0$ if $n$ is
even. Then, it follows easily from [5, (18)] that

$\displaystyle\zeta_{\mathcal{A}}(\bar{s})=\sum_{\Pi=(P_{1},\dots,P_{\ell})\in[d]}\mathcal{C}(\Pi)\zeta_{\mathcal{A}}\Big{(}\overline{s|P_{1}|}\Big{)}\cdots\zeta_{\mathcal{A}}\Big{(}\overline{s|P_{\ell}|}\Big{)}.$

The proposition follows from (14) immediately. ∎

######  Example 3.15.

There are many ways to partition $6$ elements, say $\\{a_{1},\dots,a_{6}\\}$
into odd parts: one way to get $(\\{1\\}^{6})$, $\binom{6}{5}$ ways to obtain
$(1,5)$ (e.g., $\\{a_{2}\\},\\{a_{1},a_{3},\dots,a_{6}\\}$), $\binom{6}{3}/2$
ways to obtain $(3,3)$, and $\binom{6}{3}$ ways to obtain $(1,1,1,3)$. Hence,

$\zeta_{\mathcal{A}}(\\{\bar{1}\\}^{6})=\tfrac{4}{45}{\texttt{q}}_{2}^{6}+\tfrac{3}{4}{\texttt{q}}_{2}\beta_{5}+\tfrac{1}{8}\beta_{3}^{2}+\tfrac{2}{3}{\texttt{q}}_{2}^{3}\beta_{3}$

when using the formula in (14). We would like to point out that the term
$3q_{p}B_{p-5}/20$ (corresponding to the second term
$\tfrac{3}{4}{\texttt{q}}_{2}\beta_{5}$ on the right-hand side above) was
accidentally dropped from the right-hand side of [19, (36)].

One may compare the next corollary with the well-known result that
$\zeta_{\mathcal{A}}(\\{1\\}^{d})=0$ for all $d\in\mathbb{N}$ (see, e.g., [27,
Theorem 8.5.1]).

######  Proposition 3.16.

For all $d\in\mathbb{N}$, we have

$T_{\mathcal{A}}(\\{1\\}^{2d})=0.$

###### Proof.

Taking ${\boldsymbol{\sl{s}}}=(\\{1\\}^{2d-1})$ in Theorem 3.4 yields the
proposition at once. ∎

We now derive the symmetric MTV version of Proposition 3.16.

######  Proposition 3.17.

For all $d\in\mathbb{N}$, we have

$T_{\shuffle}^{\mathcal{S}}(\\{1\\}^{2d})=0.$

###### Proof.

For any $\ell\in\mathbb{N}$, we have the relation for the regularized value
(see, e.g., [7, Section 2])

$\int_{0}^{\varepsilon}\bigg{(}\frac{dt}{1-t^{2}}\bigg{)}^{\ell}=\frac{1}{\ell!}\bigg{(}\int_{0}^{\varepsilon}\frac{dt}{1-t^{2}}\bigg{)}^{\ell}=\frac{1}{\ell!}\bigg{(}\frac{1}{2}\int_{0}^{\varepsilon}\bigg{(}\frac{dt}{1-t}+\frac{dt}{1+t}\bigg{)}\bigg{)}^{\ell},$

which implies that

$T_{\shuffle}(\\{1\\}^{\ell})=\frac{1}{\ell!2^{\ell}}\Big{(}\zeta_{\shuffle}(1)+\log
2\Big{)}^{\ell}.$

According to the definition,

$T_{\shuffle}^{\mathcal{S}}(\\{1\\}^{2d})=\sum_{i=0}^{2d}(-1)^{i}T_{\shuffle}(\\{1\\}^{i})T_{\shuffle}(\\{1\\}^{2d-i})=\frac{1}{2^{2d}}\sum_{i=0}^{2d}\frac{(-1)^{i}}{\ell!(2d-\ell)!}\Big{(}\zeta_{\shuffle}(1)+\log
2\Big{)}^{2d}=0$

as desired. ∎

By conducting extensive numerical experiments, we found that the following
relations must be valid.

######  Conjecture 3.18.

For all odd $w\in\mathbb{N}$, we have

$T_{\mathcal{A}}(\\{1\\}^{w})=-S_{\mathcal{A}}(\\{1\\}^{w})=\frac{2^{w-1}-1}{2^{2w-2}}\beta_{w},\quad
T_{\shuffle}^{\mathcal{S}}(\\{1\\}^{w})=-S_{\shuffle}^{\mathcal{S}}(\\{1\\}^{w})=\frac{2^{w-1}-1}{2^{2w-2}}\zeta(w).$

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
$T_{\mathcal{A}}(\\{1\\}^{w})=T_{\shuffle}^{\mathcal{S}}(\\{1\\}^{w})=0$ 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 $w\in\mathbb{N}$, there are rational numbers
$c_{j}\in\mathbb{Q}$, $1\leq j\leq w/2$, such that

$S_{\mathcal{A}}(\\{1\\}^{w})=\sum_{j=1}^{w/2}c_{j}S_{\mathcal{A}}(j,w-j),\quad
S_{\shuffle}^{\mathcal{S}}(\\{1\\}^{w})=\sum_{j=1}^{w/2}c_{j}S_{\shuffle}^{\mathcal{S}}(j,w-j).$

Moreover, $S_{\mathcal{A}}(j,w-j)$ and $1\leq j\leq w/2$ are
$\mathbb{Q}$-linearly independent, and $S_{\shuffle}^{\mathcal{S}}(j,w-j)$ and
$1\leq j\leq w/2$ are $\mathbb{Q}$-linearly independent.

Note that $S_{\mathcal{A}}(j,w-j)\in\mathcal{A}$ when
$S_{\shuffle}^{\mathcal{S}}(j,w-j)$ are all real numbers.

##  4 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
$({\boldsymbol{\sl{s}}},{\boldsymbol{\sigma}})\in\mathbb{N}^{d}\times\\{\pm
1\\}^{d}$ , we have defined the finite alternating multiple $T$-values as

$T({\boldsymbol{\sl{s}}};{\boldsymbol{\sigma}}):=\bigg{(}\sum_{\begin{subarray}{c}p>n_{1}>\dots>n_{d}>0\\\
n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-d+j-1)/2}}{n_{j}^{s_{j}}}\bigg{)}_{p\in\mathcal{P}}\in\mathcal{A}.$

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.

######  Proposition 4.1 (Reversal relations of finite alternating MTVs).

Let ${\boldsymbol{\sl{s}}}\in\mathbb{N}^{d}$ for some even $d\in\mathbb{N}$.
Then,

$\displaystyle
T_{\mathcal{A}}(\overleftarrow{{\boldsymbol{\sl{s}}}},\overleftarrow{{\boldsymbol{\sigma}}})=$
$\displaystyle\,({\sigma}_{1},\dots,{\sigma}_{d})^{(p-1-d)/2}(-1)^{|{\boldsymbol{\sl{s}}}|}T_{\mathcal{A}}({\boldsymbol{\sl{s}}},{\boldsymbol{\sigma}}),$
(17)

where the element
$(-1)^{(p-1-d)/2}=\big{(}(-1)^{(p-1-d)/2}\pmod{p}\big{)}_{3\leq\in\mathcal{P}}\in\mathcal{A}$.

###### Proof.

Let $p$ be an odd prime. Then, by changing the indices $n_{j}\to p-n_{j}$, we
obtain

$\displaystyle T_{p}({\boldsymbol{\sl{s}}},{\boldsymbol{\sigma}}):=$
$\displaystyle\,\sum_{\begin{subarray}{c}p>n_{1}>\dots>n_{d}>0\\\ n_{j}\equiv
d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-d+j-1)/2}}{n_{j}^{s_{j}}}$
$\displaystyle\equiv$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}\sum_{\begin{subarray}{c}p>n_{d}>\dots>n_{1}>0\\\
p-n_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\sigma}_{j}^{(n_{j}-p+d-j+1)/2}}{n_{j}^{s_{j}}}\pmod{p}.$

Let $t_{j}=s_{d+1-j}$, ${\varepsilon}_{j}={\sigma}_{d+1-j}$, and
$k_{j}=n_{d+1-j}$. Then, by changing the indices, we obtain $j\to d+1-j$
(since $d$ is even)

$\displaystyle T_{p}({\boldsymbol{\sl{s}}},{\boldsymbol{\sigma}})\equiv$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}\sum_{\begin{subarray}{c}p>n_{1}>\dots>n_{d}>\\\
p-k_{j}\equiv j{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\varepsilon}_{j}^{(k_{j}-p+j)/2}}{k_{j}^{t_{j}}}$
$\displaystyle\equiv$
$\displaystyle\,(-1)^{|{\boldsymbol{\sl{s}}}|}\sum_{\begin{subarray}{c}p>n_{1}>\dots>n_{d}>\\\
p-k_{j}\equiv j{\ (\rm{mod}\
2)}\end{subarray}}({\sigma}_{1}\cdots{\sigma}_{d})^{(d-p+1)/2}\prod_{j=1}^{d}\frac{{\varepsilon}_{j}^{(k_{j}-d+j-1)/2}}{k_{j}^{t_{j}}}$
$\displaystyle\equiv$
$\displaystyle\,({\sigma}_{1}\cdots{\sigma}_{d})^{(d-p+1)/2}(-1)^{|{\boldsymbol{\sl{s}}}|}\sum_{\begin{subarray}{c}p>k_{1}>\dots>k_{d}>\\\
k_{j}\equiv d-j+1{\ (\rm{mod}\
2)}\end{subarray}}\prod_{j=1}^{d}\frac{{\varepsilon}_{j}^{(k_{j}-d+j-1)/2}}{k_{j}^{t_{j}}}$
$\displaystyle\equiv$
$\displaystyle\,({\sigma}_{1}\cdots{\sigma}_{d})^{(d-p+1)/2}(-1)^{|{\boldsymbol{\sl{s}}}|}T_{p}({\boldsymbol{\sl{t}}},{\boldsymbol{\varepsilon}})$
$\displaystyle\equiv$
$\displaystyle\,({\sigma}_{1}\cdots{\sigma}_{d})^{(d-p+1)/2}(-1)^{|{\boldsymbol{\sl{s}}}|}T_{p}(\overleftarrow{{\boldsymbol{\sl{s}}}},\overleftarrow{{\boldsymbol{\sigma}}})$

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

$\displaystyle S_{\mathcal{A}}(\bar{1})=-{\emph{{q}}}_{2}/2,\quad
T_{\mathcal{A}}(\bar{1})=\Big{(}(-1)^{\tfrac{p-1}{2}}q_{2}(p)/2\pmod{p}\Big{)}_{p>2}\in\mathcal{A}.$

###### Proof.

Recall that

$S_{p}(1):=\sum_{p>k>0,2|k}\frac{1}{k},\quad
S_{p}(\bar{1}):=\sum_{p>k>0,2|k}\frac{(-1)^{k/2}}{k}.$

According to [18, Theorem 3.2], we see that

$S_{p}(1)+S_{p}(\bar{1})=\sum_{p>k>0,2|k}\bigg{(}\frac{1}{k}+\frac{(-1)^{k/2}}{k}\bigg{)}=\sum_{p>k>0,4|k}\frac{2}{k}\equiv-\frac{3}{2}q_{p}(2)\pmod{p}.$

Since $S_{p}(1)=\zeta_{p}(\bar{1})/2=-q_{p}(2)$, we immediately see that
$S_{\mathcal{A}}(\bar{1})=-{\texttt{q}}_{2}/2$. By taking the reversal, we
obtain

$\displaystyle T_{p}(\bar{1})=\sum_{p>k>0,2\nmid k}\frac{(-1)^{(k-1)/2}}{k}=$
$\displaystyle\,\sum_{p>k>0,2|k}\frac{(-1)^{(p-k-1)/2}}{p-k}$
$\displaystyle\equiv$
$\displaystyle\,-(-1)^{\tfrac{p-1}{2}}S_{p}(\bar{1})\equiv(-1)^{\tfrac{p-1}{2}}\frac{q_{p}(2)}{2}\pmod{p},$

as desired. ∎

As we analyzed in [27, p. 239], there is overwhelming evidence that
${\texttt{q}}_{2}\neq 0$ in $\mathcal{A}$. In [16, Theorem 1], Silverman even
showed that if abc-conjecture holds, then

$\Big{|}\Big{\\{}p\leq X:q_{2}(p)\neq
0\pmod{p}\Big{\\}}\Big{|}=O(\log(X))\quad\text{as}\quad X\to\infty.$

In we are sure the following conjecture is true.

######  Conjecture 4.3.

For every pair of positive integers $m>a>0$, $\gcd(m,a)=1$, there are
infinitely many primes $p\equiv a\pmod{m}$ such that $q_{2}(p)\not\equiv
0\pmod{p}$.

######  Theorem 4.4.

If Conjecture 4.3 holds for $m=4$, then $T_{\mathcal{A}}(1)$ and
$T_{\mathcal{A}}(\bar{1})$ are $\mathbb{Q}$-linearly independent.

###### Proof.

If $c_{1}T_{\mathcal{A}}(1)+c_{2}T_{\mathcal{A}}(\bar{1})=0$ in $\mathcal{A}$
for some $c_{1},c_{2}\in\mathbb{Q}$, then, according to Proposition 4.2, we
see that $(c_{1}+c_{2})q_{p}(2)\equiv 0\pmod{p}$ for infinitely many primes
$p\equiv 1\pmod{4}$. If Conjecture 4.3 holds the form $m=4$, then
$c_{1}+c_{2}\equiv 0\pmod{p}$ for infinitely many primes $p\equiv 1\pmod{4}$.
This would force $c_{1}+c_{2}=0$. A similar consideration for primes $p\equiv
3\pmod{4}$ implies that $c_{1}-c_{2}=0$. Hence, we must have $c_{1}=c_{2}=0$,
which shows that $T_{\mathcal{A}}(1)$ and $T_{\mathcal{A}}(\bar{1})$ are
$\mathbb{Q}$-linearly independent. ∎

Define the _finite Catalan’s constant_ as

$G_{\mathcal{A}}:=\Big{(}\frac{E_{p-3}}{2}\Big{)}_{3<p\in\mathcal{P}}\in\mathcal{A}.$

######  Proposition 4.5.

Let $\mathsf{FAMT}_{w}$ be the vector space generated by finite alternating
MTVs over $\mathbb{Q}$. We have the following generating sets of
$\mathsf{FAMT}_{w}$ for $w<3$:

$\displaystyle\mathsf{FAMT}_{1}=\langle{\emph{{q}}}_{2},(-1)^{p^{\prime}}{\emph{{q}}}_{2}\rangle,\quad\mathsf{FAMT}_{2}=\langle
G_{\mathcal{A}},(-1)^{p^{\prime}}G_{\mathcal{A}}\rangle.$

###### Proof.

The $w=1$ case is trivial. For $w=2$, we already know
$T_{\mathcal{A}}(1,1)=T_{\mathcal{A}}(2)=0$ from Theorem 3.13. Let
${\texttt{a}}={\texttt{y}}_{0}$, ${\texttt{b}}={\texttt{y}}_{1}$, and
${\texttt{c}}={\texttt{y}}_{-1}$ in the rest of the proof. For alternating
values, we first have the linear shuffle relation

$T_{\mathcal{A}}({\texttt{b}}\shuffle{\texttt{c}})=-T_{\mathcal{A}}({\texttt{b}}{\texttt{c}})\Rightarrow
2T_{\mathcal{A}}({\texttt{b}}{\texttt{c}})+T_{\mathcal{A}}({\texttt{c}}{\texttt{b}})\Rightarrow
2T_{\mathcal{A}}(1,\bar{1})+T_{\mathcal{A}}(\bar{1},\bar{1})=0.$

By using complicated computation (see Proposition 4.4 of our arxiv paper
2402.08160 and notice (7)), we have the additional relation

$T_{\mathcal{A}}(\bar{2})=G_{\mathcal{A}}=-2T_{\mathcal{A}}(1,\bar{1}).$

Then, from the reversal relation (17), we easily see that
$T_{\mathcal{A}}(\bar{1},1)=-(-1)^{p^{\prime}}T_{\mathcal{A}}(1,\bar{1}).$
This completes the proof of the proposition. ∎

##  5 Dimensions of $\mathsf{FMT}$ and $\mathsf{AMT}$

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,

$\displaystyle T(\bar{2},\bar{3})=$
$\displaystyle\,\sum_{n_{1}>n_{2}>0}\frac{(-1)^{n_{1}-2}(-1)^{n_{2}-1}}{(2n_{1}-2)^{2}(2n_{2}-1)^{3}}$
$\displaystyle=$
$\displaystyle\,\sum_{k_{1}>k_{2}>0}\frac{i^{k_{1}}(1+(-1)^{k_{1}})i^{k_{2}-1}(1-(-1)^{k_{2}})}{k_{1}^{2}k_{2}^{3}}$
$\displaystyle=$
$\displaystyle\,-i\Big{(}Li_{2,3}(i,i)+Li_{2,3}(i,-i)-Li_{2,3}(-i,i)-Li_{2,3}(-i,-i)\Big{)}.$

The caveat is that we need to extend our scalars to $\mathbb{Q}[i]$ in
general. At the end of [17], we observed that
$\dim_{\mathbb{Q}}\mathsf{FCMZ}^{4}_{w}\leq 2^{w}$ for all $w\geq 1$, where
$\mathsf{FCMZ}^{4}_{w}$ is the space spanned by all colored MVZ of level 4 and
weight $w$ over $\mathbb{Q}$. By the following, we expect that the

$\dim_{\mathbb{Q}}\mathsf{FAMT}_{w}\leq\dim_{\mathbb{Q}}\mathsf{FAM}_{w}\leq
2^{w},$

where $\mathsf{FAM}$ 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. (1)

Suppose all elements in $S$ have the same weight. If they are linearly
independent over $\mathbb{Q}$, then they are algebraically independent over
$\mathbb{Q}$.

  2. (2)

If the weights of the values in $S$ are all different, then the values are
linearly independent over $\mathbb{Q}$ (but, of course, may not be
algebraically independent over $\mathbb{Q}$).

  3. (3)

If there is only one nonzero element in $S$, then it is transcendental over
$\mathbb{Q}$.

For example, we expect that $\zeta(n)$’s are not only irrational but are also
transcendental for all $n\geq 2$. We also expect that ${\texttt{q}}_{2}$ and
$\beta_{k}$ are transcendental for all odd $k\geq 3$ and are all algebraically
independent over $\mathbb{Q}$

Recall that $\mathsf{MT}_{w}$ (resp. $\mathsf{FMT}_{w}$) is the
$\mathbb{Q}$-vector space generated by MTVs (resp. finite MTVs) of weight $w$.
Similarly, we denote by $\mathsf{AMT}$ (resp. $\mathsf{FAMT}$) 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 $\mathsf{FMT}_{w}$ and $\mathsf{FAMT}_{w}$. 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.

w | 0 | 1 | _2_ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathsf{FMT}_{w}$ | 0 | 1 | 0 | 1 | 2 | 3 | 3 | 6 | 9 | 15 | 17 | 32 | 44 | 76
$\mathsf{MT}_{w}$ | 1 | 0 | 1 | 1 | 2 | 2 | 4 | 5 | 9 | 10 | 19 | 23 | 42 | 49
$\mathsf{FAMT}_{w}$ | 0 | 2 | 2 | 6 | 12 | 20 | 40 | 76 |  |  |  |  |  |
$\mathsf{AMT}_{w}$ | 0 | 1 | 2 | 4 | 7 | 13 | 24 | 44 | 81 |  |  |  |  |
$\mathsf{FAM}_{w}$ | 0 | 1 | 2 | 4 | 8 | 16 |  |  |  |  |  |  |  |

With strong numerical support, Xu and the author conjecture that
$\\{\dim_{\mathbb{Q}}\mathsf{AMT}_{w}\\}_{w\geq 1}$ form the tribonacci
sequence (see [22, Conjecture 5.2]). For MTVs, Kaneko and Tsumura conjecture
in [10] that, for all $k\geq 1$

$\displaystyle\dim_{\mathbb{Q}}\mathsf{MT}_{2k}=\dim_{\mathbb{Q}}\mathsf{MT}_{2k-1}+\dim_{\mathbb{Q}}\mathsf{MT}_{2k-2}.$

From numerical computation, we can formulate its finite analog as follows:

######  Conjecture 5.2.

For all $k\geq 1$,

$\displaystyle\dim_{\mathbb{Q}}\mathsf{FMT}_{2k+1}=\dim_{\mathbb{Q}}\mathsf{FMT}_{2k}+\dim_{\mathbb{Q}}\mathsf{FMT}_{2k-1}.$

##  6 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
$\mathbb{Q}$-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.

Acknowledgment. This research is supported by the Jacobs Prize from The
Bishop’s School. The author thanks the referees for their detailed comments
which has helped improve the clarity of the paper.

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