On rapidly convergent series for the Riemann Zeta-values via the modular relation

2002 ◽  
Vol 72 (1) ◽  
pp. 187-206 ◽  
Author(s):  
Shigeru Kanemitsu ◽  
Yoshio Tanigawa ◽  
Masami Yoshimoto
Keyword(s):  
Convergent Series ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Modular Relation

scholarly journals Decomposition of products of Riemann zeta values

2015 ◽  
Vol 150 ◽  
pp. 1-20 ◽  
Author(s):  
Chan-Liang Chung ◽  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Zeta Values ◽  
Riemann Zeta

Sum formulas of multiple zeta values with selected arguments

2018 ◽  
Vol 14 (10) ◽  
pp. 2617-2630
Author(s):  
Minking Eie ◽  
Tung-Yang Lee
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Positive Integers

For positive integers [Formula: see text] with [Formula: see text] and [Formula: see text], let [Formula: see text] be the sum of multiple zeta values of depth [Formula: see text] and weight [Formula: see text] with arguments [Formula: see text] or [Formula: see text], i.e. [Formula: see text] In this paper, we are going to evaluate [Formula: see text]. As an application, we produce the stuffle relations from [Formula: see text] identical Riemann zeta values [Formula: see text] as well as [Formula: see text] identical Riemann zeta values [Formula: see text] and [Formula: see text].

scholarly journals Algorithms for Some Euler-Type Identities for Multiple Zeta Values

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shifeng Ding ◽  
Weijun Liu
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
Convergent Series ◽  
Multiple Zeta Values ◽  
Double Product ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

Multiple zeta values are the numbers defined by the convergent seriesζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), wheres1,s2,…,skare positive integers withs1>1. Fork≤n, letE(2n,k)be the sum of all multiple zeta values with even arguments whose weight is2nand whose depth isk. The well-known resultE(2n,2)=3ζ(2n)/4was extended toE(2n,3)andE(2n,4)by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbersE(2n,k)and then gave a direct formula forE(2n,k)for arbitraryk≤n. In this paper we apply a technique introduced by Granville to present an algorithm to calculateE(2n,k)and prove that the direct formula can also be deduced from Eisenstein's double product.

scholarly journals On multiple zeta values of even arguments

2017 ◽  
Vol 13 (03) ◽  
pp. 705-716 ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Bernoulli Number ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta

For [Formula: see text], let [Formula: see text] be the sum of all multiple zeta values with even arguments whose weight is [Formula: see text] and whose depth is [Formula: see text]. Of course [Formula: see text] is the value [Formula: see text] of the Riemann zeta function at [Formula: see text], and it is well known that [Formula: see text]. Recently Shen and Cai gave formulas for [Formula: see text] and [Formula: see text] in terms of [Formula: see text] and [Formula: see text]. We give two formulas for [Formula: see text], both valid for arbitrary [Formula: see text], one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers [Formula: see text] and for the analogous numbers [Formula: see text] defined using multiple zeta-star values of even arguments.

scholarly journals WEIGHTED SUMS WITH TWO PARAMETERS OF MULTIPLE ZETA VALUES AND THEIR FORMULAS

2012 ◽  
Vol 08 (08) ◽  
pp. 1903-1921 ◽  
Author(s):  
TOMOYA MACHIDE
Keyword(s):  
Weighted Sums ◽  
Multiple Zeta Values ◽  
Linear Combinations ◽  
Fixed Weight ◽  
Rational Polynomial ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula ◽  
Riemann Zeta ◽  
Two Parameters

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 formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coefficients in the rational polynomial ring generated by the two parameters, and obtain some identities for weighted sums of multiple zeta values.

scholarly journals Convolution of Riemann zeta-values

2005 ◽  
Vol 57 (4) ◽  
pp. 1167-1177 ◽  
Author(s):  
Shigeru KANEMITSU ◽  
Yoshio TANIGAWA ◽  
Masami YOSHIMOTO
Keyword(s):  
Zeta Values ◽  
Riemann Zeta

scholarly journals RATIONAL APPROXIMATIONS TO VALUES OF BELL POLYNOMIALS AT POINTS INVOLVING EULER’S CONSTANT AND ZETA VALUES

2012 ◽  
Vol 92 (1) ◽  
pp. 71-98
Author(s):  
KH. HESSAMI PILEHROOD ◽  
T. HESSAMI PILEHROOD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Bell Polynomials ◽  
Rational Approximations ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Image Position ◽  
Euler's Constant

AbstractIn this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

scholarly journals On evaluation formulas for double L-values

2004 ◽  
Vol 70 (2) ◽  
pp. 213-221 ◽  
Author(s):  
Hirofumi Tsumura
Keyword(s):  
Special Cases ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Positive Integers

In this paper, we give some evaluation formulas for the values of double L-series of Tornheim's type, in terms of the Dirichlet L-values and the Riemann zeta values at positive integers. As special cases, these give the formulas for double L-values given by Terhune.

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