tailieunhanh - Evaluation of Euler-like sums via Hurwitz zeta values

In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. | Turk J Math (2017) 41: 1640 – 1655 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Evaluation of Euler-like sums via Hurwitz zeta values ˙ 1,∗, Istv´ ˝ 2 , Mehmet CENKCI˙ 3 Ayhan DIL an MEZO Department of Mathematics, Akdeniz University, Antalya Turkey 2 Department of Mathematics, Nanjing University of Information Science and Technology, Pukou, Nanjing, Jiangsu, PR China 3 Department of Mathematics, Akdeniz University, Antalya Turkey 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. In light of this notion we evaluate some particular values of Euler sums in terms of odd zeta values. We also consider the noninteger property and some arithmetical aspects of this unified extension. Key words: Harmonic numbers, hyperharmonic numbers, generalized harmonic numbers, Euler sums, multiple zeta functions, Riemann zeta function, Hurwitz zeta function. 1. Introduction The nth harmonic number is defined by Hn := n ∑ 1 k (n ∈ N := {1, 2, 3, . . .}) , k=1 where the empty sum H0 is conventionally understood to be zero. Harmonic numbers are a longstanding subject of study and they are significant in various branches of analysis and number theory. These numbers are closely related to the Riemann zeta function defined by ζ (s) = ∞ ∑ 1 ns n=1 (Re (s) > 1) and appear in the expressions of miscellaneous special functions. Among many other generalizations we are interested in two famous generalizations of harmonic numbers, namely generalized harmonic numbers and hyperharmonic numbers. Generalized harmonic numbers: The generalized n th harmonic number of order m is defined by the nth partial .