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.

ON A WEIGHTED SUM OF MULTIPLE -VALUES OF FIXED WEIGHT AND DEPTH

2021 ◽  
pp. 1-8
Author(s):  
YOSHIHIRO TAKEYAMA
Keyword(s):  
Differential Equation ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Multiple Zeta Value ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Pure Mathematics ◽  
Zeta Values

AbstractThe multipleT-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in:Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multipleT-values of fixed weight and depth is given in terms of the multipleT-values of depth one by solving a differential equation of Heun type.

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 Multiple zeta values, poly-Bernoulli numbers, and related zeta functions

1999 ◽  
Vol 153 ◽  
pp. 189-209 ◽  
Author(s):  
Tsuneo Arakawa ◽  
Masanobu Kaneko
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Multiple Zeta Values ◽  
Special Values ◽  
Multiple Zeta ◽  
Zeta Values

AbstractWe study the function and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.

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

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.

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