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.

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 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 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.

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 On Vectorized Weighted Sum Formulas of Multiple Zeta Values

2016 ◽  
Vol 20 (2) ◽  
pp. 243-261
Author(s):  
Chan-Liang Chung ◽  
Yao Lin Ong
Keyword(s):  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values
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