On some weighted sum formulas involving general multiple zeta-type series

2019 ◽  
Vol 205 ◽  
pp. 124-147
Author(s):  
Marian Genčev
Keyword(s):  
Weighted Sum ◽  
Type Series ◽  
Multiple Zeta

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

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.

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