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We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall's identity for the generalized hypergeometric function 5 F 4 . Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

We introduce a new Lie-algebraic approach to explicitly construct the motivic coaction and single-valued map of multiple polylogarithms in any number of variables. In both cases, the appearance of multiple zeta values is controlled by conjugating generating series of polylogarithms with Lie-algebra generators associated with odd zeta values. Our reformulation of earlier constructions of coactions and single-valued polylogarithms preserves choices of fibration bases, exposes the correlation between multiple zeta values of different depths and paves the way for generalizations beyond genus zero. [ABSTRACT FROM AUTHOR]

Yamamoto's integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto's integral. We determine the value of infinite series of the form: $ \sum\limits_{m_1,\ldots,m_n,\ell_1,\ldots,\ell_k\geq 1}\frac{H_{m_1}^{(a_1)}\cdots H_{m_n}^{(a_n)}} {m_1^{b_1}\cdots m_n^{b_n}\ell_1^{c_1}\cdots\ell_k^{c_k} \binom{m_1+\cdots+m_n+\ell_1+\cdots+\ell_k}{\ell_k}}, $ in terms of a finite sum of multiple zeta values, for positive integers $ a_1, \ldots, a_n, b_1, \ldots, b_n, c_1, \ldots, c_k $.

In this paper, we solve the open problem posed by Kuba by expressing ∑ j , k ≥ 1 H k (u) H j (v) H j + k (w) j r k s (j + k) t as a linear combination of multiple zeta values. These sums include Tornheim's double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. [ABSTRACT FROM AUTHOR]

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues λ (n) of shapes with n edges that are of the form λ (n) ∼ x ∑ k = 0 ∞ C k (x) n k where x is the limiting eigenvalue for n → ∞ . Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order C k (x) and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove an evaluation for the stuffle-regularised alternating multiple t value t ∗ , V (1 ¯ , ... , 1 ¯ , 1 , 1 ¯ , ... , 1 ¯) in terms of V , the regularisation parameter, log (2) , ζ (k) and β (k) . This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a zero-balanced hypergeometric function 3 F 2 , and an evaluation established by Li in an alternative approach to Zagier's evaluation of ζ (2 , ... , 2 , 3 , 2 , ... , 2) . We end with some discussion and conjectures on possible motivic applications. [ABSTRACT FROM AUTHOR]

We provide a framework for relating certain q-series defined by sums over partitions to multiple zeta values. In particular, we introduce a space of polynomial functions on partitions for which the associated q-series are q-analogues of multiple zeta values. By explicitly describing the (regularized) multiple zeta values one obtains as q → 1 , we extend previous results known in this area. Using this together with the fact that other families of functions on partitions, such as shifted symmetric functions, are elements in our space will then give relations among (q-analogues of) multiple zeta values. Conversely, we will show that relations among multiple zeta values can be 'lifted' to the world of functions on partitions, which provides new examples of functions for which the associated q-series are quasimodular. [ABSTRACT FROM AUTHOR]

The goal of the paper is twofold. First, we present an analytic method leading to a class of combinatorial identities with Bernoulli, Euler and Catalan numbers based on considering specific multiple zeta-like series and infinite products. The developed method allows us to naturally extend Hoffman’s combinatorial identity that led to the famous evaluation of the multiple zeta value ζ ({ 2 } k) in 1992. Second, we present new evaluations of two multiple zeta-like series with their consequences to combinatorial identities, and, as a by-product of our technical considerations, we establish two combinatorial identities with a trinomial coefficient and Stirling numbers respectively. [ABSTRACT FROM AUTHOR]

We construct a q-analogue of the truncated version of symmetric multiple zeta values, which satisfies the double shuffle relation. Using it, we define a q-analogue of symmetric multiple zeta values and see that it satisfies many of the same relations as symmetric multiple zeta values, which are the reversal relation and a part of the double shuffle relation and the Ohno-type relation. [ABSTRACT FROM AUTHOR]

In this paper, we define some weighted sums of the alternating multiple T-values (AMTVs) and study several duality formulas for them. Then we introduce the alternating version of the convoluted T-values and Kaneko–Tsumura ψ -function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the Q -vector space generated by the AMTVs of any fixed weight w and provide some evidence for the conjecture that their dimensions { d w } w ≥ 1 form the tribonacci sequence 1, 2, 4, 7, 13,.... [ABSTRACT FROM AUTHOR]

In this paper, we prove a duality formula for the sum of the finite/symmetric multiple zeta-star values of fixed weight, depth, and height, which was conjectured by Kaneko. This result is a generalization of "height-one duality" proved by Hoffman and an analogue of the duality formula for the sum of the multiple zeta-star values proved by Li. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problem posed by Kuba by expressing ∑j,k≥1Hk(u)Hj(v)Hj+k(w)jrks(j+k)t as a linear combination of multiple zeta values. These sums include Tornheim’s double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series.

It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals. [ABSTRACT FROM AUTHOR]

Classical multiple zeta values can be viewed as iterated integrals of the differentials d t t , d t 1 − t from 0 to 1. In this paper, we reprove Brown's theorem: For a i , b i , c i j ∈ ℤ , the iterated integral of the form ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i t i a i (1 − t i) b i ∏ i < j (t j − t i) c i j d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values of weight ≤ N if convergent. What is more, we show that if p i (t) , 1 ≤ i ≤ N , are in a ℚ [ t , 1 / t , 1 / (1 − t) ] -algebra generated by multiple polylogarithms and their dual, and if q i j (t) , 1 ≤ i < j ≤ N , are in a ℚ [ t , 1 / t ] -algebra generated by logarithm, then the iterated integral ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i p i (t i) ∏ i < j q i j (t j − t i) d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 f ∏ i t i α i (1 − t i) β i ∏ i < j (t j − t i) γ i j d t 1 ⋯ d t N (with respect to α i , β i , γ i j ) at the integral points in some product of right half complex plane are ℚ -linear combinations of multiple zeta values for any f ∈ ℚ [ t i , t i − 1 , (t i − t j) − 1 | 1 ≤ i ≤ N , 1 ≤ i < j ≤ N ]. This statement generalizes Terasoma's original result. [ABSTRACT FROM AUTHOR]

The tangent symmetry of multiple zeta values at non-positive integers is discussed from the viewpoint of a certain functional relation for the multiple zeta-functions. Further, several relationships between multiple zeta values at non-positive integers for various limiting processes are presented by using the tangent symmetry. [ABSTRACT FROM AUTHOR]

We construct a family of q-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given Q -valued solution of the extended double shuffle equations. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given Q -valued solution of the extended double shuffle equations (as q → 0 ) and multiple zeta values (as q → 1 ). In particular, they are q-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould. [ABSTRACT FROM AUTHOR]

Multiple zeta values at non-positive integers depend on its limiting process. In this paper, explicit and recurrence formulas to evaluate multiple zeta values at non-positive integers for various limiting processes are given. As applications of them, a classification of multiple zeta values for ordered limits and relationship between multiple zeta values for directional limits and ordered limits are showed. Further, an alternative proof of the conjecture raised by Akiyama, Egami and Tanigawa is also given. [ABSTRACT FROM AUTHOR]

Parametrized multiple series are generalizations of the multiple zeta values introduced by Igarashi. In this work, we completely determine all the linear relations among these parametrized multiple series. Specifically, we prove the following two statements: the linear part of the Kawashima relation for multiple zeta values can be generalized to parametrized multiple series; any linear relations among the parametrized multiple series can be written as a linear combination of the linear part of the Kawashima relation. [ABSTRACT FROM AUTHOR]

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t-modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t-modules which lack the height property. In particular, let A be the polynomial ring in one variable \theta over the finite field \mathbb {F}_q with the quotient field K. Let (E, \phi) be an iterated extension of the tensor powers of the Carlitz module defined over K. We show that the \mathbb {F}_q[t]-module E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free \mathbb {F}_q[t]-module of rank \aleph _0. [ABSTRACT FROM AUTHOR]

Based on the Connes–Kreimer Hopf algebra of rooted trees, rooted tree maps are defined as linear maps on the noncommutative polynomial algebra Q ⟨ x , y ⟩ . It is known that they induce a large class of linear relations for multiple zeta values. In this paper, we show for any rooted tree f there exists a unique polynomial in Q ⟨ x , y ⟩ that gives the image of the rooted tree map f ~ explicitly. We also characterize the antipode maps as the conjugation by the special map τ . [ABSTRACT FROM AUTHOR]

The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas. [ABSTRACT FROM AUTHOR]

Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations. [ABSTRACT FROM AUTHOR]

An interpolation function of the multiple harmonic sum is defined. Its basic properties, asymptotic behaviors and derivatives, which play fundamental roles in the study of the multiple polylogarithm and multiple zeta values, are investigated. [ABSTRACT FROM AUTHOR]

We prove that the dihedral Lie coalgebra...corresponding to bD••(G) of Goncharov (Duke Math. J. 110 (2001), 397--487) for G = {e} is the bigraded dual of the linearized double shuffle Lie algebra...of Brown (Compos. Math. 157 (2021), 529--572) whose Lie bracket is the Ihara bracket initially defined over Q(x, z), by constructing an explicit isomorphism of bigraded Lie coalgebras ... where ls is the Lie coalgebra dual (in the bigraded sense) to ls. The work leads to the equivalence between the two statements is a Lie coalgebra with respect to Goncharov's cobracket formula in Goncharov (Duke Math. J. 110 (2001), 397--487)" and "ls is preserved by the Ihara bracket". We also prove folklore results from Brown (Compos. Math. 157 (2021), 529--572) and Ihara et al. (Compos. Math. 142 (2006), 307--338) (which apparently have no written proofs in the literature) stating that for ... is graded isomorphic (dual) to the double shuffle space ... (stated in Ihara et al., Compos. Math. [ABSTRACT FROM AUTHOR]

Using the shuffle product method, we prove some generalized summation formulas for alternating multiple zeta values, a product formula of two alternating multiple zeta values of height one, and a formula of the product of n alternating zeta values. [ABSTRACT FROM AUTHOR]

For positive integers p 1 , p 2 , ... , p k , q with q ≥ 2 , we define the Euler T-sums T p 1 p 2 ⋯ p k , q as the infinite series whose general terms are products of odd harmonic numbers and negative powers of half integers. Similarly to the classical Euler sums, the Euler T-sums can be evaluated by the method of contour integrals and residue computation. Using this fact, we establish explicit formulas for some T-sums. The Euler T-sums can also be written as rational linear combinations of the Hoffman t-values, which help us show that the triple Hoffman t-values are reducible to zeta values, double zeta values and double t-values. We also define the alternating Euler T-sums and another odd variant of the Euler sums, which are called the Euler S ~ -sums here. We study these sums by the contour integrals and residue computation, and find many interesting results. In particular, we show that the triple Kaneko–Tsumura T-values of even weight can be expressed in terms of single and double T-values. [ABSTRACT FROM AUTHOR]

We extend the block filtration, defined by Brown based on the work of Charlton, to all motivic multiple zeta values, and study relations compatible with this filtration. We construct a Lie algebra describing relations among motivic multiple zeta values modulo terms of lower block degree, proving Charlton's cyclic insertion conjecture in this structure, and showing the existence of a 'block shuffle' relation, a dihedral symmetry, and differential relation. [ABSTRACT FROM AUTHOR]

The cyclic relation obtained in a study by Hirose, Murakami, and the first-named author, is a wide class of relations, which includes the well-known cyclic sum formula for multiple zeta and zeta-star values, and the derivation relation for multiple zeta values. In this paper, we present its generalization to complex variables. Our proof includes a new proof of the cyclic relation. [ABSTRACT FROM AUTHOR]

In this paper, we give an elementary account into Zagier's formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient ζ (2 n) . This formula plays an important role in proving Hoffman's conjecture which asserts that every multiple zeta value of weight k can be expressed as a Q -linear combinations of multiple zeta values of the same weight involving 2's and 3's. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values. [ABSTRACT FROM AUTHOR]

We study multiple zeta values (MZV's) over the rational function field over F q which were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we affirmatively solve a conjecture of Lara Rodriguez and Thakur which gives a full list of zeta-like MZV's of weight at most q 2 and depth 2. Further, we completely determine all zeta-like MZV's of weight at most q 2 and arbitrary depth. Our method is based on a criterion which is derived from the Anderson–Thakur motivic interpretation of MZV's and the Anderson–Brownawell–Papanikolas criterion for linear independence in positive characteristic. [ABSTRACT FROM AUTHOR]

This paper gives an application of so-called connected sums, introduced recently by Seki and Yamamoto [SY]. Special about our approach is that it proves a duality for the Schlesinger-Zudilin and the Bradley-Zhao model of qMZVs simultaneously. The latter implies the duality for MZVs and the former can be used to prove the shuffle product formula for MZVs. Furthermore, the q-Ohno relation, a generalization of Bradley-Zhao duality, is also obtained. [ABSTRACT FROM AUTHOR]

We introduce the multivariable connected sum which is a generalization of Seki–Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno's relations for multiple polylogarithms. [ABSTRACT FROM AUTHOR]