scholarly journals Multiple zeta values for coordinatewise limits at non-positive integers

2009 ◽  
Vol 136 (4) ◽  
pp. 299-317 ◽  
Author(s):  
Yoshitaka Sasaki
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

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].

Several weighted sum formulas of multiple zeta values

2017 ◽  
Vol 13 (09) ◽  
pp. 2253-2264 ◽  
Author(s):  
Minking Eie ◽  
Wen-Chin Liaw ◽  
Yao Lin Ong
Keyword(s):  
Real Number ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Explicit Evaluation ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Number Formula ◽  
Positive Integers ◽  
Special Case

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

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.

ON GENERALIZATIONS OF WEIGHTED SUM FORMULAS OF MULTIPLE ZETA VALUES

2013 ◽  
Vol 09 (05) ◽  
pp. 1185-1198 ◽  
Author(s):  
YAO LIN ONG ◽  
MINKING EIE ◽  
WEN-CHIN LIAW
Keyword(s):  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula ◽  
Positive Integers

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as [Formula: see text] where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilin's weighted sum formula.

scholarly journals THE STRONG CHOWLA–MILNOR SPACES AND A CONJECTURE OF GUN, MURTY AND RATH

2012 ◽  
Vol 08 (05) ◽  
pp. 1301-1314
Author(s):  
TAPAS CHATTERJEE
Keyword(s):  
Lower Bound ◽  
Recent Work ◽  
Number Field ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

In a recent work, Gun, Murty and Rath formulated the Strong Chowla–Milnor conjecture and defined the Strong Chowla–Milnor space. In this paper, we proved a non-trivial lower bound for the dimension of these spaces. We also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both ζ(k) and ζ(k)/πk for all odd positive integers k. Following Gun, Murty and Rath, we define generalized Zagier spaces Vp(K) for multiple zeta values over a number field K. We prove that the dimension of V4d+2(K) for d ≥ 1, is at least 2, assuming a conjecture of Gun, Murty and Rath.

scholarly journals MULTI-POLY-BERNOULLI NUMBERS AND RELATED ZETA FUNCTIONS

2017 ◽  
Vol 232 ◽  
pp. 19-54 ◽  
Author(s):  
MASANOBU KANEKO ◽  
HIROFUMI TSUMURA
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Multiple Zeta Values ◽  
Linear Combinations ◽  
Multiple Zeta ◽  
Zeta Values ◽  
The One ◽  
Positive Integers

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\unicode[STIX]{x1D709}$-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

scholarly journals A Combinatorial Identity of Multiple Zeta Values with Even Arguments

10.37236/3923 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shifeng Ding ◽  
Lihua Feng ◽  
Weijun Liu
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Combinatorial Identity ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Generating Series ◽  
Zeta Values ◽  
Positive Integers

Let $\zeta(s_1,s_2,\cdots,s_k;\alpha)$ be the multiple Hurwitz zeta function. Given two positive integers $k$ and $n$ with $k\leq n$, let $E(2n, k;\alpha)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$.  In this note we present some generating series for the numbers $E(2n,k;\alpha)$.

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