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Ohno-Wakabayashi's cyclic sum formula for multiple zeta-star values is generalized by Igarashi with one or two parameters. In this article, we give a possible answer for one of his problems about a generalization with three parameters.

There are three kinds of multiple polylogarithms; complex, finite and symmetric. The dualities for the complex and finite cases are known. In this paper, we present proofs of them via iterated integrals and its symmetric counterpart by a similar method.

In this paper, we obtain a restricted decomposition formula for interpolated multiple zeta values using t-stuffle product. We then derive a recursive formula of t-stuffle product, which also provides a route to the same formula. In both cases, combinatorial description of t-stuffle product is our basic tool. We also provide alternative proofs by mathematical induction for some of the results., Comment: 22 Pages

We study the asymptotic behavior of a multiple series of Mordell-Tornheim type and its integral analogue at x=0. Our approach is to show a relation between the multiple series and its integral analogue by using Abel's summation formula, and to deeply investigate the behavior of the integral analogue. Additionally, we establish some nontrivial relations among multiple polylogarithms by comparing two seemingly different asymptotic formulas for the integral analogue., Comment: 30pages

In this paper, we consider a discrete version of iterated integrals by the naive (equally divided) Riemann sum. In particular, basic three formulas for usual iterated integrals are discritized. Moreover, we proved cyclic sum formulas for discrete iterated integrals. They imply the cyclic sum formula for multiple polylogarithms.

Recently, Maesaka, Watanabe, and the third author discovered a phenomenon where the iterated integral expressions of multiple zeta values become discretized. In this paper, we extend their result to the case of multiple polylogarithms and provide two proofs. The first proof uses the method of connected sums, while the second employs induction based on the difference equations that discrete multiple polylogarithms satisfy. We also investigate several applications of our main result., Comment: 29 pages

In this paper, we construct generating functions of alternating sums for the Arakawa-Kaneko zeta values. From the expressions, we show alternating sum formulas for them. Based on these results, we apply the same method to other zeta values., Comment: 14 pages

We give an explicit formula for the well-known parity result for multiple zeta values as an application of the multitangent functions.

Xu, Yan and Zhao showed that in even weight, the multiple $T$ value $T(2, 1, \ldots, 1, \overline{1})$ is a polynomial in $\log(2)$, $\pi$, Riemann zeta values, and Dirichlet beta values. Based on low-weight examples, they conjectured that $\log(2)$ does not appear in the evaluation. We show that their conjecture is correct, and in fact follows largely from various earlier results of theirs. More precisely, we derive explicit formulae for $T(2, 1, \ldots, 1, \overline{1})$ in even weight and $S(2, 1, \ldots, 1, \overline{1})$ in odd weight via generating series calculations. We also resolve another conjecture of theirs on the evaluations of $T(\overline{2}, 1, \ldots, 1, \overline{1})$, $S(\overline{2}, 1, \ldots, 1, 1)$, and $S(\overline{2}, 1, \ldots, 1, \overline{1})$ in even weight, by way of calculations involving Goncharov's theory of iterated integrals and multiple polylogarithms., Comment: 22 pages

Recently, Maesaka, Seki and Watanabe discovered a surprising equality between multiple harmonic sums and certain Riemann sums which approximate the iterated integral expression of the multiple zeta values. In this paper, we describe the formula corresponding to the multiple zeta-star values and, more generally, to the Schur multiple zeta values of diagonally constant indices. We also discuss the relationship of these formulas with Hoffman's duality identity and an identity due to Kawashima., Comment: 13 pages

We present a new proof of the extended double shuffle relation for multiple zeta values which notably does not rely on the use of integrals. This proof is based on a formula recently obtained by Maesaka, Watanabe, and the author., Comment: 8 pages, to appear in Kyushu Journal of Mathematics

We give a new expression of the multiple harmonic sum, which serves as a refinement of the iterated integral expression of the multiple zeta value, and prove it using the so-called connected sum method. Based on this fact, by taking two kinds of limit operations, we obtain new proofs of both the duality for multiple zeta values and the duality for finite multiple zeta values., Comment: 11 pages

The $t$-adic symmetric multiple zeta value is a generalization of the symmetric multiple zeta value from the perspective of the Kaneko-Zagier conjecture. In this paper, we introduce a further generalization with a new parameter $s$, which we call the $(s,t)$-adic symmetric multiple zeta value. Then, the $(s,t)$-adic version of the $t$-adic double shuffle relations, duality and cyclic sum formula are established. A finite counterpart of the $(s,t)$-adic symmetric multiple zeta value is also discussed.

In this paper, we consider infinite-length versions of multiple zeta-star values. We give several explicit formulas for the infinite-length versions of multiple zeta-star values. We also discuss the analytic properties of the map from indices to the infinite-length versions of multiple zeta-star values.

Shuffle algebra has been employed to give a proof of the duality theorem for multiple zeta-star values of height one.

Using the block filtration as a realisation of the coradical filtration, we study the discrepancy between the depth filtration and the coradical filtration for motivic multiple zeta values. We construct an explicit dictionary between a certain subspace of block graded multiple zeta values and totally odd multiple zeta values and show that all expected relations in the depth graded motivic Lie algebra may be realised in the block graded Lie algebra as the kernel of an explicit map. We also discuss some connections to the uneven Broadhurst-Kreimer conjecture, and outline a possible approach., Comment: 24 pages

We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system., Comment: 46 pages

In \cite{IKLNDP23} we presented a systematic study of algebra structures of multiple zeta values in positive characteristic introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we construct algebra and Hopf algebra structures of alternating multiple zeta values introduced by Harada, extending our previous work. Our results could be considered as an analogue of those of Hoffman \cite{Hof00} and Racinet \cite{Rac02} in the classical setting. The proof is based on two new ingredients: the first one is a direct and explicit construction of the shuffle Hopf algebra structure, and the second one is the notion of horizontal maps., Comment: 37 pages. arXiv admin note: text overlap with arXiv:2301.05906

In this paper, we investigate an asymptotic behavior of the double zeta function of Euler-Zagier type for indices with large negative real parts.

Recently, Kaneko and Tsumura introduced multiple $\widetilde{T}$-values, another kind of poly-Euler numbers and the related Arakawa-Kaneko type zeta function. It is shown that each of them satisfies similar formulas to those of multiple zeta values, poly-Bernoulli numbers and the related Arakawa-Kaneko type zeta function. In this paper, we show some explicit relations for values of Kaneko-Tsumura $\lambda$ function at positive integers, including the duality type relation., Comment: 14 pages

In this paper, we give a purely algebraic proof of an identity coming directly from Euler's reflection formula for the gamma function. Our proof uses Hoffman's harmonic algebra and some binomial identities., Comment: 5 pages

We show that the level 2 case of the cyclotomic Grothendieck-Teichm\"{u}ller groups introduced by Enriquez coincides with the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$.

We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height., Comment: 10 pages

Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.

The duality relation is a basic family of linear relations for multiple zeta values. The extended double shuffle relation (EDSR) is one of the families of relations expected to generate all linear relations among multiple zeta values, but it remains unclear as to whether all duality relations can be deduced from the EDSR. In the present paper, regarding the family generated by the duality relation and the family generated by the derivation relation, an explicit characterization of their intersection is obtained. Here, the derivation relation is a specialization of the EDSR., Comment: 18 pages

Let $A_\tau$ denote the elliptic associator constructed by Enriquez, a power series in two non-commutative variables $a,b$ defined as an iterated integral of the Kronecker function $F_\tau$. We study a family of {\it Fay relations} satisfied by $A_\tau$, derived from the original Fay relation satisfied by the $F_\tau$. The Fay relations of $A_\tau$ were studied by Broedel, Matthes and Schlotterer, and determined up to non-explicit correction terms that arise from the necessity of regularizing the non-convergent integral. Here we study a reduced version $\bar{A}_\tau$ mod $2\pi i$. We recall a different construction of $\bar{A}_\tau$ in three steps, due to Matthes, Lochak and the author: first one defines the reduced {\it elliptic generating series} $\bar{E}_\tau$ which comes from the reduced Drinfeld associator $\overline{\Phi}_{KZ}$ and whose coefficients generate the same ring $\bar{R}$ as those of $\bar{A}_\tau$; then one defines $\Psi$ to be the automorphism of the free associative ring $\bar{R}\langle\langle a,b\rangle\rangle$ defined by $\Psi(a)=\bar{E}_\tau$ and $\Psi([a,b])=[a,b]$; finally one shows that the reduced elliptic associator $\bar{A}_\tau$ is equal to $\Psi\bigl({{ad(b)}\over{e^{ad(b)}-1}}(a)\bigr)$. Using this construction and mould theory and working with Lie-like versions of the elliptic generating series and associator, we prove the following results: (1) a mould satisfies the Fay relations if and only if a closely related mould satisfies the "swap circ-neutrality" relations defining the elliptic Kashiwara-Vergne Lie algebra $krv_{ell}$, (2) the reduced elliptic generating series satisfies a family of Fay relations with extremely simple correction terms coming directly from those of the Drinfeld associator, and (3) the correction terms for the Fay relations satisfied by the reduced elliptic associator can be deduced explicitly from these.

We give a result on the asymptotic behavior of the Hurwitz-Lerch multiple zeta functions near non-positive integer points by using the Apostol-Bernoulli polynomials. From this result, we can evaluate limit values at non-positive integer points.

In this paper we define a symmetric zeta function. We show that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles at some special hyperplanes. We also calculate the value of a multiple residue at one special point. For a divergent multiple series, which can be viewed as the value of the symmetric zeta function at the point $(1,1,1)$, we give a very detailed analysis on its growth and we relate it to the classical Euler constant., Comment: 28 pages

In this paper we introduce and study double tails of multiple zeta values. We show, in particular, that they satisfy certain recurrence relations and deduce from this a generalization of Euler's classical formula $\zeta(2)=3\sum_{m=1}^\infty m^{-2}\left({2m\atop m}\right)^{-1}$ to all multiple zeta values, as well as a new and very efficient algorithm for computing these values.

In the present paper, for $s,t,u \in {\mathbb{C}}$, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function $T(s,t,u)$ and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new a proof of known results on the values of $T(s,s,s)$ at non-positive integers and the location of the poles of $T(s,s,s)$. Furthermore, we prove that the function $T(s,s,s)$ can not be written by a polynomial in the form of $\sum_{k=1}^j c_k \prod_{r=1}^q \zeta^{d_{kr}} (a_{kr} s + b_{kr})$, where $a_{kr}, b_{kr}, c_k \in {\mathbb{C}}$ and $d_{kr} \in {\mathbb{Z}}_{\ge 0}$., Comment: 12 pages

For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid in $\mathcal{F}_n$-multiple zeta values for both the case where $\mathcal{F}=\mathcal{A}$ and $\mathcal{F}=\mathcal{S}$. In particular, the Bowman-Bradley type theorem and sum formulas for $\mathcal{S}_2$-multiple zeta values are proved., Comment: 27 pages, to appear in Commentarii mathematici Universitatis Sancti Pauli

In the present series of papers, we study the behavior of the r-fold zeta-function of Euler-Zagier type with identical arguments on the real line. In this first part, we consider the behavior on the interval [0,1]. Our basic tool is an "infinite" version of Newton's classical identities. We carry out numerical computations, and draw graphs for real s in [0,1], for several small values of r. Those graphs suggest various properties of the r-fold zeta-function, some of which we prove rigorously. For example, we show that the r-fold zeta-function has r asymptotes, and determine the asymptotic behavior close to those asymptotes. Until now, the existence of one real zero for r=2 has been known. Our present computations establish several new real zeros between asymptotes in the cases r=3,...,10. Moreover, on the number of real zeros, we raise a conjecture, and a formula for calculating the number of zeros on the interval [0,1] is derived., Comment: 16pages, 9figures

The Ohno relation is a well known relation in the theory of multiple zeta values. Recently, Seki and Yamamoto introduced a connector method and gave its succinct proof. On the other hand, Igarashi obtained the generalization of the Ohno relation for parametrized multiple zeta series. In this paper, we give new connectors to simplify his proof., Comment: 8 pages

Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values. Gaku Kawashima introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization of the theory of regularizations of divergent multiple zeta values to Hurwitz type multiple zeta values, and then relate it to the Kawashima function. Via this connection, we can prove a key property of the Kawashima function to obtain Kawashima's relation., Comment: 14 pages

We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two multiplicative relations, which arise naturally from comparison of the MZVs with an underlying algebraic structure. We generalise these concepts by introducing the parameter $q$ in such a way that as $q\to 1^-$ we return to the ordinary MZVs. Our special interest lies in two $q$-models recently introduced by H. Bachmann. He further conjectures that the $\mathbb{Q}$-spaces generated by these $q$-generalisations coincide. In this thesis we establish a particular case of Bachmann's conjecture., Comment: 55 pages; Master thesis