scholarly journals ALTERNATING EULER SUMS AND SPECIAL VALUES OF THE WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO

2010 ◽  
Vol 89 (3) ◽  
pp. 419-430 ◽  
Author(s):  
JIANQIANG ZHAO
Keyword(s):  
Zeta Function ◽  
Linear Combination ◽  
Lie Algebra ◽  
Special Values ◽  
Multiple Zeta Function ◽  
Euler Sums ◽  
Multiple Zeta ◽  
Special Value

AbstractWe study the Witten multiple zeta function associated with the Lie algebra $\so $. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w at least 2 is a finite ℚ-linear combination of alternating Euler sums of weight w and depth at most 2, except when the only nonzero argument is one of the two last variables, in which case ζ(w−1) is needed.

scholarly journals MULTI-POLYLOGS AT TWELFTH ROOTS OF UNITY AND SPECIAL VALUES OF WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO THE EXCEPTIONAL LIE ALGEBRA 𝔤2

2010 ◽  
Vol 09 (02) ◽  
pp. 327-337 ◽  
Author(s):  
JIANQIANG ZHAO
Keyword(s):  
Zeta Function ◽  
Lie Algebra ◽  
Roots Of Unity ◽  
Linear Combinations ◽  
Exceptional Lie Algebra ◽  
Special Values ◽  
Multiple Zeta Function ◽  
Multiple Zeta

In this note, we shall study the Witten multiple zeta function associated with the exceptional Lie algebra 𝔤2. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multi-polylogs evaluated at 12th roots of unity, except for two irregular cases.

On multiple Hurwitz zeta function of Mordell–Tornheim type

2021 ◽  
pp. 1-34
Author(s):  
Kazuhiro Onodera
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Zeta Function ◽  
Multiple Zeta ◽  
Second Derivatives ◽  
Positive Integers

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers.

Generalized harmonic numbers and Euler sums

2017 ◽  
Vol 13 (02) ◽  
pp. 513-528 ◽  
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Zeta Functions ◽  
Multiple Zeta Values ◽  
Harmonic Numbers ◽  
Special Values ◽  
Euler Sums ◽  
Summation Formulas ◽  
Multiple Zeta ◽  
Zeta Values ◽  
New Formula ◽  
Generalized Harmonic Numbers

In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa–Kaneno zeta functions and a new formula of multiple zeta values of height one as examples.

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