Finite Multiple Zeta Values Research Articles

Weighted sum formula for variants of half multiple zeta values

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 [Formula: see text]. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed.

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

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

Mordell–Tornheim Zeta Values, Their Alternating Version, and Their Finite Analogs

The purpose of this paper is two-fold. First, we consider the classical Mordell–Tornheim zeta values and their alternating version. It is well-known that these values can be expressed as rational linear combinations of multiple zeta values (MZVs) and the alternating MZVs, respectively. We show that, however, the spaces generated by the Mordell–Tornheim zeta values over the rational numbers are in general much smaller than the MZV space and the alternating MZV space, respectively, which disproves a conjecture of Bachmann, Takeyama and Tasaka. Second, we study supercongruences of some finite sums of multiple integer variables. This kind of congruences is a variation of the so called finite multiple zeta values when the moduli are primes instead of prime powers. In general, these objects can be transformed to finite analogs of the Mordell–Tornheim sums which can be reduced to multiple harmonic sums. This approach not only simplifies the proof of a few previous results but also generalizes some of them. At the end of the paper, we provide a general conjecture involving this type of sums, which is supported by strong numerical evidence.

On the refined Kaneko–Zagier conjecture for general integer indices

The refined Kaneko–Zagier conjecture claims that the algebras spanned by two kinds of “completed” finite multiple zeta values, called {widehat{mathcal {A}}}- and {widehat{mathcal {S}}}-MZVs, are isomorphic. Recently, Komori defined {widehat{mathcal {S}}}-MZVs of general integer (i.e., not necessarily positive) indices, extending the existing definition for positive indices. In view of the refined Kaneko–Zagier conjecture, Komori’s work suggests that these extended values are closely connected to {widehat{mathcal {A}}}-MZVs of general indices, which can be defined in an obvious way. In this paper, we show that the generalization of the refined Kaneko–Zagier conjecture for general integer indices is actually deduced from the conjecture for positive indices. The key ingredient is an inductive formula for {widehat{mathcal {A}}}-MZVs or {widehat{mathcal {S}}}-MZVs of indices which contain at least one non-positive entry.

On finite multiple zeta values of level two

We introduce and study a ``level two'' analogue of finite multiple zeta values. We give conjectural bases of the space of finite Euler sums as well as that of usual finite multiple zeta values in terms of these newly defined elements. A kind of ``parity result'' and certain sum formulas are also presented.

Yamamoto's Interpolation of Finite Multiple Zeta and Zeta-star Values

We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in particular, prove the cyclic sum formula, the Bowman--Bradley type formula, and the weighted sum formula. The harmonic relation, the shuffle relation, the duality relation, and the derivation relation are also presented.

Finite multiple zeta values, symmetric multiple zeta values and unified multiple zeta functions

We introduce entire multiple zeta functions, which are natural interpolations of symmetric multiple zeta values. Moreover we give further generalizations which interpolate these zeta functions, $\widehat{\mathcal{A}}$-finite multiple zeta values and $\widehat{\mathcal{S}}$-symmetric multiple zeta values. We also show that the correspondence which Kaneko--Zagier conjecture suggests holds on nonpositive integers for finite multiple zeta values and special values of these multiple zeta functions.

Finite and symmetric colored multiple zeta values and multiple harmonic q-series at roots of unity

The Kaneko–Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous works with H. Bachmann and Y. Takeyama, we proved that the finite and symmetric multiple zeta values are obtained as an ‘algebraic’ and ‘analytic’ limit at $$q\rightarrow 1$$ of certain multiple harmonic q-sums, and studied their relations in order to give partial evidence of the Kaneko–Zagier conjecture. In this paper, we start with multiple harmonic q-sums of level N, which are q-analogues of the truncated colored multiple zeta values. We introduce our finite and symmetric colored multiple zeta values as an algebraic and analytic limit of the multiple harmonic q-sums of level N and discuss a higher level (or a cyclotomic) analogue of the Kaneko–Zagier conjecture.

On variants of symmetric multiple zeta-star values and the cyclic sum formula

The t-adic symmetric multiple zeta values were defined by Jarossay, which have been studied as a real analogue of the \({\varvec{p}}\)-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko–Yamamoto’s type regularization. We also present the cyclic sum formula for t-adic symmetric multiple zeta(-star) values, which is the counterpart of that for \({\varvec{p}}\)-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our new relationship that connects the cyclic sum formula for t-adic symmetric multiple zeta-star values and that for the multiple zeta-star values.

OHNO-TYPE RELATIONS FOR CLASSICAL AND FINITE MULTIPLE ZETA-STAR VALUES

Ohno's relation is a generalization of both the sum formula and the duality formula for multiple zeta values. Oyama gave a similar relation for finite multiple zeta values, defined by Kaneko and Zagier. In this paper, we prove relations of similar nature for both multiple zeta-star values and finite multiple zeta-star values. Our proof for multiple zeta-star values uses the linear part of Kawashima's relation.

Finite and symmetric Mordell–Tornheim multiple zeta values

We introduce finite and symmetric Mordell–Tornheim type of multiple zeta values and give a new approach to the Kaneko–Zagier conjecture stating that the finite and symmetric multiple zeta values satisfy the same relations.

Quasi-derivation relations for multiple zeta values revisited

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

On a generalization of the cyclic sum formula for finite multiple zeta and zeta-star values

The cyclic sum formulas for multiple zeta and zeta-star values were respectively proved by Hoffman and Ohno, and Ohno and Wakabayashi. Kawasaki and Oyama obtained analogous formulas for finite multiple zeta and zeta-star values. In this paper, we give a generalization of Kawasaki and Oyama’s results.

Bowman-Bradley type theorem for finite multiple zeta values in $\mathcal{A}_2$

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings $\{1,3,\ldots,1,3\}$ and $\{2,\ldots,2\}$ is a rational multiple of a power of $\pi^2$. Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring $\mathcal{A}_1$ vanish. In this paper, we partially lift Saito-Wakabayashi's theorem from $\mathcal{A}_1$ to $\mathcal{A}_2$. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in $\mathcal{A}_2$ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.

Depth reductions for associators

We prove that for any associator, two specific families of coefficients of the associator can be expressed in terms of coefficients of lower depth. Combining these results to our notions of adjoint p-adic multiple zeta values and multiple harmonic values, we obtain a new point of view on the question of relating p-adic and finite multiple zeta values, and a few other application to the study of p-adic multiple zeta values via explicit formulas.

Ohno-type identities for multiple harmonic sums

We establish Ohno-type identities for multiple harmonic ($q$-)sums which generalize Hoffman's identity and Bradley's identity. Our result leads to a new proof of the Ohno-type relation for $\mathcal{A}$-finite multiple zeta values recently proved by Hirose, Imatomi, Murahara and Saito. As a further application, we give certain sum formulas for $\mathcal{A}_2$- or $\mathcal{A}_3$-finite multiple zeta values.

DERIVATION RELATION FOR FINITE MULTIPLE ZETA VALUES IN

AbstractIhara et al. proved the derivation relation for multiple zeta values. The first-named author obtained its counterpart for finite multiple zeta values in ${\mathcal{A}}$. In this paper, we present its generalization in $\widehat{{\mathcal{A}}}$.

Double Shuffle Relations for Refined Symmetric Multiple Zeta Values

Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that symmetric multiple zeta values satisfy double shuffle relations and duality relations. In this paper, we construct certain lifts of SMZVs which live in the ring generated by all multiple zeta values and $2\pi i$ as certain iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed path. We call this lifted values as refined symmetric multiple zeta values (RSMZVs). We show double shuffle relations and duality relations for RSMZVs. These relations are refinements of the double shuffle relations and the duality relations of SMZVs. Furthermore we compare RSMZVs to other variants of lifts of SMZVs. Especially, we prove that RSMZVs coincide with Bachmann-Takeyama-Tasaka's $\xi$-values.

Restricted sum formula for finite and symmetric multiple zeta values

The sum formula for finite and symmetric multiple zeta values, established by Wakabayashi and the authors, implies that if the weight and depth are fixed and the specified component is required to be more than one, then the values sum up to a rational multiple of the analogue of the Riemann zeta value. We prove that the result remains true if we further demand that the component should be more than two or that another component should also be more than one.

ON A GENERALISATION OF A RESTRICTED SUM FORMULA FOR MULTIPLE ZETA VALUES AND FINITE MULTIPLE ZETA VALUES

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.