Harmonic number sums in closed form (Q2897915)

\textit{A. Sofo} [Hacet. J. Math. Stat. 39, No. 2, 255--263 (2010; Zbl 1198.05010)] had already found some of the identities here generalized. This paper extends also an identity supplied by \textit{B. Cloitre} [{\texttt{http://mathworld.wolfram.com/HarmonicNumber.html}}], a recursion known to \textit{L. Euler} [Opera omnia (Latin) Leipzig: B. G. Teubner (1911, 1980; JFM 53.0020.03)] and other results obtained by \textit{M. Jung, Y. J. Cho} and \textit{J. Choi} [Commun. Korean Math. Soc. 19, No. 3, 545--555 (2004; Zbl 1101.11324)] and expressed in terms of Riemann \(\zeta\)- functions.NEWLINENEWLINEHence the present paper refines various earlier works, even from the same [\textit{A. Sofo}, Appl. Math. Comput. 207, No. 2, 365--372 (2009; Zbl 1175.11007), Adv. Appl. Math. 42, No. 1, 123--134 (2009; Zbl 1220.11025), Integral Transforms Spec. Funct. 20, No. 11--12, 847--857 (2009; Zbl 1242.11014)].NEWLINENEWLINEBeyond employing basic computational techniques for the summation of series described in [\textit{A. Sofo}, Computational techniques for the summation of series. New York, NY: Kluwer Academic/Plenum Publishers. (2003; Zbl 1059.65002)], in the proof the author develops the integral and closed-form representations of sums with products of harmonic numbers and cubed binomial coefficients in terms of \textit{Polygamma function} whose evaluation at rational arguments is given through formulae provided by \textit{K. S. Kölbig} [J. Comput. Appl. Math. 75, No. 1, 43--46 (1996; Zbl 0860.33002)] or by \textit{J. Choi} and \textit{D. Cvijović} [J. Phys. A, Math. Theor. 40, No. 50, 15019--15028 (2007; Zbl 1127.33002)] using Polylogarithmic and other special functions.NEWLINENEWLINEThe interest in evaluating closed form and integral representations of Euler type sums containing both harmonic numbers and powers of binomial coefficients is remarked by the author who recalls several related papers, \(e.g.\), by \textit{H. Alzer, D. Karayannakis} and \textit{H. M. Srivastava} [J. Math. Anal. Appl. 320, No. 1, 145--162 (2006; Zbl 1093.33010)], \textit{D. Borwein, J. M. Borwein} and \textit{R. Girgensohn} [Proc. Edinb. Math. Soc., II. Ser. 38, No. 2, 277--294 (1995; Zbl 0819.40003)], \textit{X. Chen} and \textit{W. Chu} [Discrete Math. 310, No. 1, 83--91 (2010; Zbl 1250.33007)], \textit{W. Chu} and \textit{A. M. Fu} [Ramanujan J. 18, No. 1, 11--31 (2009; Zbl 1175.05009)], \textit{W. Chu} and \textit{D. Zheng} [Int. J. Number Theory 5, No. 3, 429--448 (2009; Zbl 1191.05014)], \textit{P. Flajolet} and \textit{B. Salvy} [Exp. Math. 7, No. 1, 15--35 (1998; Zbl 0920.11061)], \textit{C. Krattenthaler} and \textit{K. Srinivasa Rao} [J. Comput. Appl. Math. 160, No. 1--2, 159--173 (2003; Zbl 1038.33003)] and \textit{A. Basu} [Ramanujan J. 16, No. 1, 7--24 (2008; Zbl 1155.40002)].