scholarly journals Multiple zeta values and application to the Lacunary recurrence formulas of Bernoulli numbers

2008 ◽  
Vol 96 ◽  
pp. 012212 ◽  
Author(s):  
Y-H Chen
Keyword(s):  
Bernoulli Numbers ◽  
Multiple Zeta Values ◽  
Recurrence Formulas ◽  
Multiple Zeta ◽  
Zeta Values

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 New Techniques for Worldline Integration

2021 ◽  
Author(s):  
James P. Edwards ◽  
◽  
C. Moctezuma Mata ◽  
Uwe Müller ◽  
Christian Schubert ◽  
...  
Keyword(s):  
String Theory ◽  
State Of The Art ◽  
Bernoulli Numbers ◽  
Integral Representations ◽  
Feynman Diagrams ◽  
Multiple Zeta Values ◽  
New Techniques ◽  
Large Classes ◽  
Multiple Zeta ◽  
Zeta Values

The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. We summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.

scholarly journals Families of weighted sum formulas for multiple zeta values

2015 ◽  
Vol 11 (03) ◽  
pp. 997-1025 ◽  
Author(s):  
Li Guo ◽  
Peng Lei ◽  
Jianqiang Zhao
Keyword(s):  
Bernoulli Numbers ◽  
Weighted Sum ◽  
Symmetric Polynomials ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Double Zeta ◽  
General Conjecture ◽  
Zeta Values ◽  
Sum Formula ◽  
Weight Coefficients

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 infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.

scholarly journals Explicit formulas of alternating multiple zeta star values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $

2021 ◽  
Vol 7 (1) ◽  
pp. 288-293
Author(s):  
Junjie Quan ◽  
Keyword(s):  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Recurrence Formulas ◽  
Multiple Zeta ◽  
Zeta Values

<abstract><p>In a recent paper <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>, Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.</p></abstract>

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 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.

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