Many $\mathbb{Q}$-linear relations exist between multiple zeta values, the most interesting of which are various weighted sum formulas. In this paper, we generalized these to Euler sums and some other variants of multiple zeta values by considering the generating functions of the Euler sums. Through this approach we are able to re-prove a few known formulas, confirm a conjecture of Kaneko and Tsumura on triple $T$-values, and discover many new identities.
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 $λ$ function at positive integers, including the duality type relation.
Recently, the level two analogue of multiple polylogarithm function ${\rm A}(k_1,\ldots,k_r;z)$ and Arakawa-Kaneko zeta function $ψ(k_1,\ldots,k_r;s)$ were introduced by M. Kaneko and H. Tsumura, for $k_1,\ldots,k_r \in \mathbb{Z}_{\ge 1}$ . In this paper, we investigate some of their special relations. In particular, we prove some explicit forms of ${\rm A}(k_1,\ldots,k_r;z)$ and $ψ(k_1,\ldots,k_r;s)$. Also, we introduce a level $m$ anlogue of the Arakawa-Kaneko zeta functions.
By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and establish several symmetric extensions of the Kaneko-Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and the…
Kaneko and Tsumura introduced a new kind of multiple zeta functions $η(k_1,\ldots,k_r;s_1,\ldots,s_r)$. This is an analytic function of complex variables $s_1,\ldots,s_r$, while $k_1,\ldots,k_r$ are non-positive integer parameters. In this paper, we first extend this function to an analytic function $η(s'_1,\ldots,s'_r;s_1,\ldots,s_r)$ of $2r$ complex variables. Then we investigate its special values at positive integers. In particular, we prove some linear relations among these $η$-values and the multiple zeta values $ζ(k_1,\ldots,k_r)$ of Euler-Zagier type.
In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study …
For these two decades, the Arakawa-Kaneko zeta function has been studied actively. Recently Kaneko and Tsumura constructed its variants from the viewpoint of poly-Bernoulli numbers. In this paper, we generalize their zeta functions of Arakawa-Kaneko type to those with indices in which positive and negative integers are mixed. We show that values of these functions at positive integers can be expressed in terms of the multiple Hurwitz zeta star values.
A Unified Approach to Formal Arakawa-Kaneko Type Zeta Values via Iterated Integrals
College of Teacher Education, Quzhou University（衢州学院教师教育学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；Department of Mathematics, The Bishop’s School（圣公会中学数学系）
Multiple $T$-values, a variant of multiple zeta values of level two, were introduced and studied by Kaneko and Tsumura. This paper will introduce iterated log-tangent integrals and discuss their relations with multiple $T$-values. We will use ideas from the author's previous work on multiple zeta values and iterated log-sine integrals to do so.
In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the linear parametric Euler $T$-sums. Furthermore, by applying the results, we obtain explicit formulae for the Hoffman's (alternating) double $t$-values and Kaneko-Tsumura's (alternating) double $T$-values.
This article introduces an algebra of functions in one variable $c$ defined by iterated integrals of two specific differential forms depending on $c$, where the product is the shuffle product. This algebra can be seen as a common deformation of multiple zeta values and of Kaneko-Tsumura's recent multiple $T$-values. The first few graded dimensions, assuming that a grading by the weight does hold, are computed.
For positive integers $p_1,p_2,\ldots,p_k,q$ with $q>1$, we define the Euler $T$-sum $T_{p_1p_2\cdots p_k,q}$ as the sum of those terms of the usual infinite series for the classical Euler sum $S_{p_1p_2\cdots p_k,q}$ with odd denominators. Like the Euler sums, the Euler $T$-sums can be evaluated according to the Contour integral and residue theorem. Using this fact, we obtain explicit formulas for Euler $T$-sums with repeated arguments analogous to those known for Euler sums. Euler $T$-sums can be written as rational linear combinations of the Hoffman $t$-values. Using known results for Hoffm…
Recently, a new kind of multiple zeta value level two $T({\bf k})$ (which is called multiple $T$-values) was introduced and studied by Kaneko and Tsumura. In this paper, we define a kind of alternating version of multiple $T$-values, and study several duality formulas of weighted sum formulas about alternating multiple $T$-values by using the methods of iterated integral representations and series representations. Some special values of alternating multiple $T$-values can also be obtained.
In this paper, we define some weighted sums of the alternating multiple $T$-values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. 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 $\{d_w\}_{w\ge 1}$ form the tribonacci sequence 1, 2, 4, 7, 13, ....
In this paper, we study the alternating Euler $T$-sums and $§$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $§$-sums, from which, the parity theorems of Hoffman's double and triple $t$-values and Kaneko-Tsumura's double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $§$-sums are expressible in…
The multiple T-value, which is a variant of multiple zeta value of level two, is introduced by Kaneko and Tsumura. We show that the generating function of a weighted sum of the multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.
We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $μ_2$. We find a basis for the depth-graded motivic double zeta values relative to $μ_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $μ_2$ of odd weight. As an application of our main results, we prove Kaneko and Tasaka's conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka's conjecture in depth three. At last we formulate a conjecture which is related to sum odd m…
