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Explicit evaluation of Euler and related sums. (English) Zbl 1115.11052
Summary: Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 33B15 Gamma, beta and polygamma functions 33E20 Other functions defined by series and integrals 11M35 Hurwitz and Lerch zeta functions 11M41 Other Dirichlet series and zeta functions 33C20 Generalized hypergeometric series, $${}_pF_q$$
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