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.

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 Some relations deduced from regularized double shuffle relations of multiple zeta values

2020 ◽  
pp. 1-56
Author(s):  
Zhonghua Li ◽  
Chen Qin
Keyword(s):  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of Guo and Xie, some evaluation formulas with even arguments and the restricted sum formulas of Hoffman and their generalizations.

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 Weighted sum formula for multiple zeta values

2009 ◽  
Vol 129 (11) ◽  
pp. 2747-2765 ◽  
Author(s):  
Li Guo ◽  
Bingyong Xie
Keyword(s):  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula

scholarly journals ON MULTIPLE ZETA VALUES OF EXTREMAL HEIGHT

2015 ◽  
Vol 93 (2) ◽  
pp. 186-193 ◽  
Author(s):  
MASANOBU KANEKO ◽  
MIKA SAKATA
Keyword(s):  
Explicit Formula ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Maximal Height ◽  
Zeta Values ◽  
Sum Formula

We give three identities involving multiple zeta values of height one and of maximal height: an explicit formula for the height-one multiple zeta values, a regularised sum formula and a sum formula for the multiple zeta values of maximal height.

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 Sum formula for finite multiple zeta values

2015 ◽  
Vol 67 (3) ◽  
pp. 1069-1076 ◽  
Author(s):  
Shingo SAITO ◽  
Noriko WAKABAYASHI
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula

ON A GENERALISATION OF A RESTRICTED SUM FORMULA FOR MULTIPLE ZETA VALUES AND FINITE MULTIPLE ZETA VALUES

2019 ◽  
Vol 101 (1) ◽  
pp. 23-34
Author(s):  
HIDEKI MURAHARA ◽  
TAKUYA MURAKAMI
Keyword(s):  
Linear Relation ◽  
Analogous Result ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Natural Generalisation ◽  
Sum Formula

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

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