منابع مشابه
Evaluation of Euler-zagier Sums
We present a simple method for evaluation of multiple Euler sums in terms of single and double zeta values. 2000 Mathematics Subject Classification. 11M99, 40B05.
متن کاملEvaluation of Triple Euler Sums
Let a, b, c be positive integers and define the so-called triple, double and single Euler sums by
متن کاملExperimental Evaluation of Euler Sums
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...
متن کاملConsecutive Evaluation of Euler Sums
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...
متن کاملExplicit evaluation of Euler sums
In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t 13), h (s; t) can be evaluated in terms of Riemann-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we...
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ژورنال
عنوان ژورنال: Mathematical Sciences
سال: 2012
ISSN: 2251-7456
DOI: 10.1186/2251-7456-6-10