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

Read more

Summary

INTRODUCTION

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

MAIN RESULTS
Hyperharmonic and generalized harmonic numbers via derivative operator
Harmonic numbers via difference operator
Hyperharmonic numbers via difference operator
Fibonacci numbers via difference operator
Hyperbolic functions via difference operator

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.