Some Properties and Generating Functions of Generalized Harmonic Numbers

Abstract

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers.

Some Sums Involving Generalized Harmonic and r-Derangement Numbers

“…Recently, using generating functions, there are some works including generalized harmonic, 𝑟 −derangement and special numbers by authors [5,6,7,9,10,11,14,15,16,17,18,20,23,24,25,26,27,31]. At the same time, many studies have been carried out on the degenerate states of these numbers [28,29,30,32,33].…”

Several recurrence relations and identities on generalized derangement numbers

“…For further investigations concerning with generalized harmonic numbers, the readers may consult with [1,2,4,7,10,11,27,28] and references cited therein.…”

Some extensions for the several combinatorial identities

“…Mortenson [9, p. 990, Lemma 3.1] gave the following identities: [1,[6][7][8][10][11][12][13]. W. Chu studied some finite combinatorial identities involving the harmonic numbers by applying the partial fraction decomposition [2][3][4].…”