A restricted sum formula among multiple zeta values
type: Item
instance of: scholarly article
title: A restricted sum formula among multiple zeta values
zbMATH Open document ID: 1183.11053
author: Minking Eie
author: Wen-Chin Liaw
author: Yao Lin Ong
published in: Journal of Number Theory
publication date: 2009-04-09T00:00:00Z
review text: For positive integers \(\alpha_1,\alpha_2,\dots,\alpha_r\) with \(\alpha_r\geq 2\), the multiple zeta value or \(r\)-fold Euler sum is defined as \[ \zeta(\alpha):= \zeta(\alpha_1,\dots, \alpha_r)= \sum_{1\leq k_1<\cdots< k_r} k^{-\alpha_1}_1\cdots k^{-\alpha_r}_r. \] There is a celebrated sum formula among multiple zeta values as \[ \sum_{|\alpha|= m}\zeta(\alpha_1,\alpha_2,\dots, \alpha_r+ 1)= \zeta(m+1), \] where \(\alpha_1,\dots, \alpha_r\) range over all positive integers with \(|\alpha|= \alpha_1+\cdots+ \alpha_r= m\) in the summation. In the present paper, the authors prove that for all positive integers \(m\) and \(q\) with \(m\geq q\), and a nonnegative integer \(p\), \[ \sum_{|\alpha|= m}\zeta(\{1\}^p,\alpha_1,\alpha_2,\dots, \alpha_q+1)= \sum_{|c|= p+q} \zeta(c_1,c_2, \dots, c_{p+1}+(m-q)+1). \] Here \(\{1\}^p\) denotes \(p\) repetitions of \(1\). When \(p=0\) and \(q=r\), this is precisely the sum formula. Such a formula can be used to determine explicitly some multiple zeta values of lower weights more precisely than the sum formula.
Mathematics Subject Classification ID: 11M32
Mathematics Subject Classification ID: 33E20
zbMATH DE Number: 5542340
zbMATH Keywords: multiple zeta value
zbMATH Keywords: sum formula
zbMATH Keywords: Drinfeld integral
reviewed by: Jürgen Hinz
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.07.012
OpenAlex ID: W1998307257
cites work: Q3995284
cites work: Explicit evaluation of Euler sums
cites work: THIRTY-TWO GOLDBACH VARIATIONS
cites work: Evaluation of triple Euler sums
cites work: Multiple polylogarithms: A brief survey
cites work: On the Evaluation of Euler Sums
cites work: Explicit evaluations of extended Euler sums
cites work: Euler sums with Dirichlet characters
cites work: Double Euler sums on Hurwitz zeta functions
cites work: ON EVALUATION OF GENERALIZED EULER SUMS OF EVEN WEIGHT
cites work: A short proof for the sum formula and its generalization
cites work: Q4213356
cites work: Multiple harmonic series
cites work: Sums of triple harmonic series
cites work: Triple sums and the Riemann zeta function
cites work: A generalization of the duality and sum formulas on the multiple zeta values
cites work: EXPLICIT EVALUATION OF TRIPLE EULER SUMS
DOI: 10.1016/J.JNT.2008.07.012
Recommended article: Restricted sum formula of multiple zeta values
Recommended article: Another expression of the restricted sum formula of multiple zeta values
Recommended article: Restricted sum formula for finite and symmetric multiple zeta values
Recommended article: On restricted sum formulas for multiple zeta values with even arguments
Recommended article: A Restricted Sum Formula for a q-Analogue of Multiple Zeta Values
Recommended article: Sum formula for finite multiple zeta values
Recommended article: Some restricted sum formulas for double zeta values
Recommended article: ON A GENERALISATION OF A RESTRICTED SUM FORMULA FOR MULTIPLE ZETA VALUES AND FINITE MULTIPLE ZETA VALUES
Recommended article: A sixfold restricted sum formula for multiple zeta values at even integers
Recommended article: Identities among restricted sums of multiple zeta values
description: scientific article; zbMATH DE number 5542340