# zbMATH — the first resource for mathematics

Evaluation of Euler-like sums via Hurwitz zeta values. (English) Zbl 1424.11130
Summary: 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.

##### MSC:
 11M32 Multiple Dirichlet series and zeta functions and multizeta values 40B05 Multiple sequences and series 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M35 Hurwitz and Lerch zeta functions
Full Text:
##### References:
 [1] Ait-Amrane R, Belbachir H. Non-integerness of class of hyperharmonic numbers. Ann Math Inform 2010; 37: 7-11. · Zbl 1224.11033 [2] Ait-Amrane R, Belbachir H. Are the hyperharmonics integral? A partial answer via the small inter-vals containing primes. C R Math Acad Sci Paris 2011; 349: 115-117. · Zbl 1226.11031 [3] Alkan E. Approximation by special values of harmonic zeta function and log-sine integrals. Commun Number Theory Phys 2013; 7: 515-550. · Zbl 1312.11070 [4] Bailey DH, Borwein JM, Girgensohn R. Explicit evaluation of Euler sums. Proc Edinburgh Math Soc 1995; 38: 277-294. · Zbl 0819.40003 [5] Berndt BC. Ramanujan’s Notebooks, Part I. New York, NY, USA: Springer-Verlag, 1985. · Zbl 0555.10001 [6] Borwein JM, Girgensohn R. Evaluation of triple Euler sums. Electron J Combin 1996; 3: R23. · Zbl 0884.40005 [7] Bowman D, Bradley DM. Multiple polylogarithms: a brief survey. Contemp Math 2001; 291: 71-92. · Zbl 0998.33013 [8] Cereceda JL. An introduction to hyperharmonic numbers (classroom note). International Journal of Mathematical Education in Science and Technology 2015; 46-3: 461-469. · Zbl 1318.97004 [9] Conway JH, Guy RK. The Book of Numbers. New York, NY, USA: Springer-Verlag, 1996. [10] Dil A, Boyadzhiev KN. Euler sums of hyperharmonic numbers. J Number Theory 2015; 147: 490-498. · Zbl 1311.11019 [11] Dil A, Mez˝o I. A symmetric algorithm for hyperharmonic and Fibonacci numbers. Appl Math Comput 2008; 206: 942-951. · Zbl 1200.65104 [12] Dil A, Mez˝o I. Hyperharmonic series involving Hurwitz zeta function. J Number Theory 2010; 130: 360-369. · Zbl 1225.11032 [13] Flajolet P, Salvy B. Euler sums and contour integral representations. Experiment Math 1998; 7-1: 15-35. · Zbl 0920.11061 [14] G¨oral H, Sertba¸s DC. Almost all hyperharmonic numbers are not integers. J Number Theory 2017; 171: 495-526. · Zbl 1396.11050 [15] Ireland K, Rosen M. A Classical Introduction to Modern Number Theory. 2nd ed. New York, NY, USA: SpringerVerlag, 1990. · Zbl 0712.11001 [16] Kamano K. Dirichlet series associated with hyperharmonic numbers. Memoirs of the Osaka Institute of Technology 2011; 56: 11-15. [17] Koblitz N. p -Adic Numbers, p -Adic Analysis, and Zeta-Functions. 2nd ed. New York, NY, USA: Springer-Verlag, 1984. [18] Li ZH. On harmonic sums and alternating Euler sums. ArXiv preprint 2010; 1012.5192v3. [19] Mez˝o I. About the non-integer property of hyperharmonic numbers. Ann Univ Sci Budapest Sect Math 2007; 50: 1-8. [20] Niven I, Zuckerman H, Montgomery H. An Introduction to the Theory of Numbers. 5th ed. New York, NY, USA: Wiley 1991. · Zbl 0742.11001 [21] Theisinger L. Bemerkung ¨uber die harmonische Reihe. Monat Math 1915; 26: 132-134. · JFM 45.0419.01 [22] Zhao J. Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values. Hackensack, NJ, USA: World Scientific, 2016. · Zbl 1367.11002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.