相关论文: Integral Evaluation of Odd Euler Sums, Multiple $t…
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…
The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic…
In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all…
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.…
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in…
We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions…
In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums.…
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta…
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic…
Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star…
We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many…
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…
For $k\leq n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\geq 2$. More precisely, $E(mn,k)=\sum_{|\boldsymbol{\alpha}|=n}\zeta(m\alpha_1,m\alpha_2,\ldots,…
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the…
For a composition $I$ whose first part exceeds 1, we can define the multiple $t$-value $t(I)$ as the sum of all the terms in the series for the multiple zeta value $\zeta(I)$ whose denominators are odd. In this paper we show that if $I$ is…
In this paper, we study some Euler-Ap\'ery-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and…
The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.
We prove that the sum of multiple zeta-star values over all indices inserted two 2's into the string $(\underbrace{3,1, ..., 3,1}_{2n})$ is evaluated to a rational multiple of powers of $\pi^2$. We also establish certain conjectures on…
A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum…
Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain…