scholarly journals Zeta Value Identities From Iterated Integrals With Additional Factors

2019 ◽  
Vol 11 (5) ◽  
pp. 40
Author(s):  
Chan-Liang Chung ◽  
Minking Eie
Keyword(s):  
Integral Representation ◽  
Linear Combination ◽  
Multiple Zeta Values ◽  
Iterated Integrals ◽  
Multiple Zeta Value ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Sum Formula

A multiple zeta value can always be represented by its Drinfel’d integral. If we add some factors appeared in the integrand of the integral representation of the multiple zeta value, it would still represent a linear combination of multiple zeta values, but the depths and weights may decrease. In this paper, we shall investigate some of multiple zeta values obtained from Drinfel’d integral with additional factors aforementioned and study a class of deformation of multiple zeta values. Results are then obtained as analogues or generalizations of the sum formula of multiple zeta values.

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.

scholarly journals Hoffman’s conjectural identity

2019 ◽  
Vol 15 (01) ◽  
pp. 167-171 ◽  
Author(s):  
Minoru Hirose ◽  
Nobuo Sato
Keyword(s):  
Multiple Zeta Values ◽  
Iterated Integrals ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Special Case

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on [Formula: see text] for our proof.

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

scholarly journals Iterated integrals on products of one variable multiple polylogarithms

2020 ◽  
Vol 16 (10) ◽  
pp. 2167-2186
Author(s):  
Jiangtao Li
Keyword(s):  
Multiple Zeta Values ◽  
Iterated Integrals ◽  
Multiple Zeta ◽  
Multiple Polylogarithms ◽  
Series Representations ◽  
Zeta Values

In this paper, we show that the iterated integrals on products of one variable multiple polylogarithms from [Formula: see text] to [Formula: see text] are actually in the algebra of multiple zeta values if they are convergent. In the divergent case, we define the regularized iterated integrals from [Formula: see text] to [Formula: see text]. By the same method, we show that the regularized iterated integrals are also in the algebra of multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals.

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.

scholarly journals Duality/sum formulas for iterated integrals and their application to multiple zeta values

2019 ◽  
Vol 195 ◽  
pp. 72-83 ◽  
Author(s):  
Minoru Hirose ◽  
Kohei Iwaki ◽  
Nobuo Sato ◽  
Koji Tasaka
Keyword(s):  
Multiple Zeta Values ◽  
Iterated Integrals ◽  
Multiple Zeta ◽  
Zeta Values

scholarly journals Cyclotomic analogues of finite multiple zeta values

2018 ◽  
Vol 154 (12) ◽  
pp. 2701-2721 ◽  
Author(s):  
Henrik Bachmann ◽  
Yoshihiro Takeyama ◽  
Koji Tasaka
Keyword(s):  
Vector Space ◽  
Primitive Root ◽  
Large Degree ◽  
Multiple Zeta Values ◽  
Linear Relations ◽  
Multiple Zeta Value ◽  
Root Of Unity ◽  
Multiple Zeta ◽  
Zeta Values

We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

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