Applications of derivative and difference operators on some sequences

TL;DR

This paper explores identities and properties of harmonic and hyperharmonic numbers using difference and derivative operators, introducing negative-ordered hyperharmonic numbers and their representations.

Contribution

It provides new identities and representations for harmonic and hyperharmonic numbers through difference and derivative operators, including the concept of negative-ordered hyperharmonic numbers.

Findings

Derived nonlinear recurrence relations for hyperharmonic numbers.

Expressed harmonic and hyperharmonic numbers as derivatives of binomial coefficients.

Introduced and characterized negative-ordered hyperharmonic numbers.

Abstract

In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number 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 number is defined and its alternative representations are given.