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Euler sums of generalized harmonic numbers and connected extensions

Can Mümün (Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey), mcan@akdeniz.edu.tr
Kargın Levent (Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey), lkargin@akdeniz.edu.tr
Dil Ayhan (Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey), adil@akdeniz.edu.tr
Soylu Gültekin (Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey), gsoylu@akdeniz.edu.tr

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ζH(p,q)(r) = ∞Xn=1 H(p,q)n/nr in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values.

Keywords: Harmonic numbers, Hyperharmonic numbers, Generalized harmonic numbers, Euler sums, Riemann zeta function