scholarly journals Algorithms for Some Euler-Type Identities for Multiple Zeta Values

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shifeng Ding ◽  
Weijun Liu
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
Convergent Series ◽  
Multiple Zeta Values ◽  
Double Product ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Positive Integers

Multiple zeta values are the numbers defined by the convergent seriesζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), wheres1,s2,…,skare positive integers withs1>1. Fork≤n, letE(2n,k)be the sum of all multiple zeta values with even arguments whose weight is2nand whose depth isk. The well-known resultE(2n,2)=3ζ(2n)/4was extended toE(2n,3)andE(2n,4)by Z. Shen and T. Cai. Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbersE(2n,k)and then gave a direct formula forE(2n,k)for arbitraryk≤n. In this paper we apply a technique introduced by Granville to present an algorithm to calculateE(2n,k)and prove that the direct formula can also be deduced from Eisenstein's double product.

Sum formulas of multiple zeta values with selected arguments

2018 ◽  
Vol 14 (10) ◽  
pp. 2617-2630
Author(s):  
Minking Eie ◽  
Tung-Yang Lee
Keyword(s):  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Positive Integers

For positive integers [Formula: see text] with [Formula: see text] and [Formula: see text], let [Formula: see text] be the sum of multiple zeta values of depth [Formula: see text] and weight [Formula: see text] with arguments [Formula: see text] or [Formula: see text], i.e. [Formula: see text] In this paper, we are going to evaluate [Formula: see text]. As an application, we produce the stuffle relations from [Formula: see text] identical Riemann zeta values [Formula: see text] as well as [Formula: see text] identical Riemann zeta values [Formula: see text] and [Formula: see text].

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 Some relations of interpolated multiple zeta values

2017 ◽  
Vol 28 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Zhonghua Li ◽  
Chen Qin
Keyword(s):  
Generating Function ◽  
Hypergeometric Functions ◽  
Multiple Zeta Values ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values

In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.

scholarly journals A Character on the Quasi-Symmetric Functions coming from Multiple Zeta Values

10.37236/821 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Mirror Symmetry ◽  
Symmetric Functions ◽  
Euler Class ◽  
Complex Manifolds ◽  
Multiple Zeta Values ◽  
Almost Complex Manifolds ◽  
Euler Constant ◽  
Multiple Zeta ◽  
Almost Complex ◽  
Zeta Values

We define a homomorphism $\zeta$ from the algebra of quasi-symmetric functions to the reals which involves the Euler constant and multiple zeta values. Besides advancing the study of multiple zeta values, the homomorphism $\zeta$ appears in connection with two Hirzebruch genera of almost complex manifolds: the $\Gamma$-genus (related to mirror symmetry) and the $\hat{\Gamma}$-genus (related to an $S^1$-equivariant Euler class). We decompose $\zeta$ into its even and odd factors in the sense of Aguiar, Bergeron, and Sottille, and demonstrate the usefulness of this decomposition in computing $\zeta$ on the subalgebra of symmetric functions (which suffices for computations of the $\Gamma$- and $\hat{\Gamma}$-genera).

Zeta Mahler measures, multiple zeta values and L-values

2015 ◽  
Vol 11 (07) ◽  
pp. 2239-2246
Author(s):  
Yoshitaka Sasaki
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Mahler Measure ◽  
Multiple Zeta Values ◽  
Explicit Formulas ◽  
Multiple Zeta ◽  
Zeta Values

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

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.

Sign in / Sign up

Export Citation Format

Share Document