Further summation formulae related to generalized harmonic numbers

By De-Yin Zheng. 2007

74 citations recorded by the source corpus.

Abstract By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ ( n ) , Trans. Amer. Math. Soc. 347 (1995) 1391–1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15–25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51–70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103–118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ ( m ) with m = 5 , 6 , 7 , including several known ones as examples.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/0b81f067fba4ef49958e3bb3bce0dcf504d23490

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Citations in the corpus

Harmonic number identities via hypergeometric series and Bell polynomials (2012, background): “[13] W. Chu and D.-Y. Zheng, Infinite series with harmonic numbers and central binomial coefficients, Int.”
A family of shifted harmonic sums (2015, background): “The works presented in (for example) [3,7,8,10,12,16,17,19,21,23,25,27,29,30] (see also [2,9] as well as the references cited in these works) investigate various closed-form representations of binomial sums and zeta functions by the use of the Beta function and by means of certain summation theorems”
An odd variant of multiple zeta values (2016, methodology): “ to Eq. (4.1) and t(3,1)+t(2,2)+2t(2,1,1) = π2 8 log2 2 = t(2)log2 2, which follows immedidately from formulas listed in the Appendix. Calculations of the same kind were carried further by D-Y. Zheng [28]: see in particular Examples 3.8 and 3.9. We have checked that Zheng’s equations are consistent ”
Harmonic number identities via the Newton–Andrews method (2014, background): “Additionally, for some other recent works on harmonic number identities, the readers are referred to [3–7,9–11,15,22–28,30,42,47].”