Multinomial coefficients, multiple zeta values, Euler sums and series of powers of the Hurwitz zeta function

2020 ◽  
Vol 62 (2) ◽  
pp. 227-245
Author(s):  
Michał Kijaczko
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Zeta Values ◽  
Euler Sums ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Multinomial Coefficients

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 $E(2n,k;\alpha)$.

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 Structural properties of multiple zeta values

2021 ◽  
pp. 1-25
Author(s):  
Tanay Wakhare ◽  
Christophe Vignat
Keyword(s):  
Zeta Function ◽  
Structural Properties ◽  
Zeta Functions ◽  
Complex Numbers ◽  
Arbitrary Sequence ◽  
Multiple Zeta Values ◽  
Quasisymmetric Functions ◽  
Multiple Zeta ◽  
Zeta Values

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built from an arbitrary sequence of nonzero complex numbers. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

scholarly journals On an analog of the Arakawa-Kaneko zeta function and relations of some multiple zeta values

2018 ◽  
Vol 42 (2) ◽  
pp. 259-294
Author(s):  
Ryota Umezawa
Keyword(s):  
Zeta Function ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values

scholarly journals Multiple zeta values and Euler sums

2017 ◽  
Vol 177 ◽  
pp. 443-478 ◽  
Author(s):  
Ce Xu
Keyword(s):  
Multiple Zeta Values ◽  
Euler Sums ◽  
Multiple Zeta ◽  
Zeta Values

Generalized harmonic numbers and Euler sums

2017 ◽  
Vol 13 (02) ◽  
pp. 513-528 ◽  
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Zeta Functions ◽  
Multiple Zeta Values ◽  
Harmonic Numbers ◽  
Special Values ◽  
Euler Sums ◽  
Summation Formulas ◽  
Multiple Zeta ◽  
Zeta Values ◽  
New Formula ◽  
Generalized Harmonic Numbers

In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa–Kaneno zeta functions and a new formula of multiple zeta values of height one as examples.

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 Explicit formulas of Euler sums via multiple zeta values

2020 ◽  
Vol 101 ◽  
pp. 109-127 ◽  
Author(s):  
Ce Xu ◽  
Weiping Wang
Keyword(s):  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Euler Sums ◽  
Multiple Zeta ◽  
Zeta Values
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