Asymptotic analysis and special values of generalised multiple zeta functions

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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Multiple zeta values, poly-Bernoulli numbers, and related zeta functions

Nagoya Mathematical Journal ◽

10.1017/s0027763000006954 ◽

1999 ◽

Vol 153 ◽

pp. 189-209 ◽

Cited By ~ 58

Author(s):

Tsuneo Arakawa ◽

Masanobu Kaneko

Keyword(s):

Zeta Functions ◽

Bernoulli Numbers ◽

Multiple Zeta Values ◽

Special Values ◽

Multiple Zeta ◽

Zeta Values

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.

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Values of multiple zeta functions with polynomial denominators at non-positive integers

International Journal of Mathematics ◽

10.1142/s0129167x21500385 ◽

2021 ◽

pp. 2150038

Author(s):

Driss Essouabri ◽

Kohji Matsumoto

Keyword(s):

Positive Integer ◽

Explicit Expression ◽

Zeta Functions ◽

Summation Formula ◽

Bernoulli Numbers ◽

Explicit Formulas ◽

Integer Points ◽

Special Values ◽

Multiple Zeta ◽

Positive Integers

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.

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