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Identities between harmonic, hyperharmonic and Daehee numbers.

Seog-Hoon Rim1, Taekyun Kim2, Sung-Soo Pyo3

  • 11Department of Mathematics Education, Kyungpook National University, Taegu, South Korea.

Journal of Inequalities and Applications
|August 24, 2018
PubMed
Summary

This study reveals new identities connecting hyperharmonic, Daehee, and derangement numbers. Researchers derived nonlinear differential equations from hyperharmonic number generating functions to establish these novel number theory relationships.

Keywords:
Daehee numbersDifferential equationHyperharmonic numbers

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Area of Science:

  • Number Theory
  • Combinatorics
  • Discrete Mathematics

Background:

  • Hyperharmonic numbers, Daehee numbers, and derangement numbers are significant combinatorial sequences.
  • Understanding relationships between these number sequences can lead to new mathematical insights.
  • Generating functions are powerful tools for studying combinatorial sequences.

Purpose of the Study:

  • To establish novel identities connecting hyperharmonic, Daehee, and derangement numbers.
  • To derive and analyze nonlinear differential equations associated with hyperharmonic numbers.
  • To explore new identities involving hyperharmonic and Daehee numbers using derived differential equations.

Main Methods:

  • Deriving identities between hyperharmonic, Daehee, and derangement numbers.
  • Utilizing the generating function of hyperharmonic numbers.
  • Formulating and solving nonlinear differential equations.
  • Applying differential equations to discover new number-theoretic identities.

Main Results:

  • Presentation of new identities involving hyperharmonic, Daehee, and derangement numbers.
  • Derivation of nonlinear differential equations from the generating function of hyperharmonic numbers.
  • Establishment of further identities incorporating hyperharmonic and Daehee numbers through the use of these differential equations.

Conclusions:

  • The study successfully established new relationships between key number sequences in combinatorics.
  • The derived differential equations provide a novel approach to uncovering identities in number theory.
  • This work contributes to a deeper understanding of hyperharmonic and Daehee numbers and their interconnections.