Rocky Mountain Journal of Mathematics

A decomposition of $\zeta(2n+3)$ into sums of multiple zeta values

Minking Eie, Wen-Chin Liaw, and Yao Lin Ong

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Abstract

We evaluate the multiple zeta value ζ ( 1 , { 2 } n + 1 ) or its dual ζ ( { 2 } n , 3 ) . When n is even, along with stuffle relations already available, it is enough to evaluate all multiple zeta values of the form ζ ( { 2 } a , 3 , { 2 } b ) with a + b = n . Furthermore, we obtain a decomposition for 2 ζ ( 2 n + 3 ) as

ζ ( 3 , { 2 } n ) + ζ ( { 2 } n , 3 ) + r = 1 n | α | = n + 1 ζ ( 1 , 2 α 1 , 2 α 2 , , 2 α r ) ,

which also can be used to evaluate ζ ( { 2 } n , 3 ) when n is even.

Article information

Source
Rocky Mountain J. Math., Volume 50, Number 2 (2020), 551-558.

Dates
Received: 12 June 2019
Revised: 3 November 2019
Accepted: 4 November 2019
First available in Project Euclid: 29 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1590739289

Digital Object Identifier
doi:10.1216/rmj.2020.50.551

Mathematical Reviews number (MathSciNet)
MR4104393

Zentralblatt MATH identifier
07210978

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 33E20: Other functions defined by series and integrals 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)

Keywords
multiple zeta value multiple zeta-star value modified Bell polynomial Bernoulli polynomial

Citation

Eie, Minking; Liaw, Wen-Chin; Ong, Yao Lin. A decomposition of $\zeta(2n+3)$ into sums of multiple zeta values. Rocky Mountain J. Math. 50 (2020), no. 2, 551--558. doi:10.1216/rmj.2020.50.551. https://projecteuclid.org/euclid.rmjm/1590739289


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