Abstract
In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.
Highlights
Harmonic numbers are longstanding subject of study and they are significant in various branches of analysis and number theory
Among many other generalizations we are interested in two important generalizations of these numbers, namely generalized harmonic numbers and hyperharmonic numbers
For m > 1, Hn(m) is the n-th partial sum of the Riemann zeta function ζ (m). Hyperharmonic numbers are another important generalization of harmonic numbers
Summary
Harmonic numbers are longstanding subject of study and they are significant in various branches of analysis and number theory. For m > 1, Hn(m) is the n-th partial sum of the Riemann zeta function ζ (m) Hyperharmonic numbers are another important generalization of harmonic numbers. Hyperharmonic numbers are closely related to analytic number theory (see [2, 5, 11, 13, 19]), discrete mathematics and combinatorial analysis (see [3, 8, 10, 12]) These numbers have an expression in terms of binomial coefficients and harmonic numbers [10, 18, 19]:. In the Appendix the reader can find two tables of summation formulas related to harmonic, hyperharmonic and generalized harmonic numbers These formulas are applications of the results that we obtained in Subsection 2.2
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