General order Euler sums with multiple argument
نویسندگان
چکیده
منابع مشابه
On the Multiple Sums of Bernoulli, Euler and Genocchi Polynomials
We introduce and investigate the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials by means of a suitable theirs generating polynomials. We establish several interesting properties of these polynomials. Also, we gave some propositions two theorems and one corollary.
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Let a, b, c be positive integers and define the so-called triple, double and single Euler sums by
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In response to a letter from Goldbach, Euler considered sums of the form ∞ ∑ k=1 ( 1 + 1 2m + · · ·+ 1 km ) (k + 1)−n for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler’s results were extended to a significantly larger class of sums of this type, including sums with alternating...
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is the Riemann zeta function. The numbers S(r ,p) for p+r odd were explicitly evaluated for the first time in [3]. As commented in [3], there is strong evidence that forp+r even S(r ,p) can be evaluated only in some particular cases: S(p,p), S(2,4), S(4,2). A powerful method for evaluation of Euler sums, based on the residue theorem, was presented in [7]. The purpose of this note is to describe...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2018
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2017.12.006