Asymptotic analysis and special values of generalised multiple zeta functions

2012 ◽  
Vol 97 ◽  
pp. 179-190
Author(s):  
M. Zakrzewski
2017 ◽  
Vol 13 (02) ◽  
pp. 513-528 ◽  
Author(s):  
Kwang-Wu Chen

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.


1999 ◽  
Vol 153 ◽  
pp. 189-209 ◽  
Author(s):  
Tsuneo Arakawa ◽  
Masanobu Kaneko

AbstractWe study the function and show that the poly-Bernoulli numbers introduced in our previous paper are expressed as special values at negative arguments of certain combinations of these functions. As a consequence of our study, we obtain a series of relations among multiple zeta values.


2021 ◽  
pp. 2150038
Author(s):  
Driss Essouabri ◽  
Kohji Matsumoto

We study rather general multiple zeta functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some “trivial zeros” exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler’s theorem, and a Raabe-type lemma due to Friedman and Pereira.


2014 ◽  
pp. 185-203
Author(s):  
M. Ram Murty ◽  
Purusottam Rath
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