Certain summation formulas involving harmonic numbers and generalized harmonic numbers
Section snippets
Introduction and preliminaries
The generalized harmonic numbers of order s are defined by (cf. Graham et al. [14]; see also [1] and [23, p. 156])andare the harmonic numbers. Here and denote the set of positive integers and the set of complex numbers, respectively, and we assume , and . We also introduce the generalized harmonic functions defined bywhich reduces to .
The Riemann
The Stirling numbers s(n, k) of the first kind
We begin by recalling the Stirling numbers s(n, k) of the first kind defined by the generating functions:andThe following recurrence relations are satisfied by s(n, k):It is not difficult to see also thatandThe
Series associated with generalized harmonic numbers
By making use of the univariate series expansion of classical hypergeometric formulas, Shen [20], and Choi and Srivastava [4], [5] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulas have been systematically derived by Chu [7] who developed fully this approach to the multivariate case (see also [8], [25]).
Here we present further interesting identities about certain finite or infinite series associated with harmonic numbers and
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).
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