Harmonic number identities via hypergeometric series and Bell polynomials

Abstract From the Kummer 2 F 1-summation theorem, the Dixon–Kummer 4 F 3-summation theorem and the Dougall–Dixon 5 F 4-summation theorem, we establish, by means of the Bell polynomials, three general formulas related to the generalized harmonic numbers and the Riemann zeta function. Based on these three general formulas, we further find series of harmonic number identities. Some of these identities involve both finite summation and infinite series, so that we can determinate the explicit expressions of numerous infinite series. In particular, we show that several interesting analogues of the Euler sums can be evaluated. Keywords: harmonic numbershypergeometric seriesBell polynomialsRiemann zeta functioncombinatorial identitiesEuler sums Acknowledgements The second author is supported by the National Natural Science Foundation of China under Grant 11001243 and the Science Foundation of Zhejiang Sci-Tech University (ZSTU) under Grant 1013817-Y.