Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

TL;DR

This paper explores properties of degenerate harmonic and higher-order harmonic numbers, linking them to degenerate zeta functions, and provides new identities and representations for related infinite sums.

Contribution

It introduces and studies degenerate harmonic numbers and their connection to degenerate zeta functions, deriving new identities and sum representations.

Findings

Infinite sums of degenerate higher-order harmonic numbers expressed via degenerate zeta functions

Representation of sums involving products of degenerate harmonic numbers using degenerate Hurwitz zeta function

Establishment of identities connecting degenerate harmonic numbers with special functions

Abstract

The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.