Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

Abstract

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijović, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear. MSC:05A15

Harmonic numbers associated with inversion numbers in terms of determinants

“…In [15], the exponential generating function of hyperharmonic numbers is given. In [16], it is shown that the sum of the series formed by hyperharmonic numbers can be expressed in terms of the Riemann zeta function.…”

Some Properties and Generating Functions of Generalized Harmonic Numbers

“…The already mentioned use of the umbral-like formalism has allowed for the framing of the theory of harmonic numbers within an algebraic context. Some of the points raised in ( [5][6][7]) have been reconsidered, made rigorous, and generalized by means of different technical frameworks in successive research ( [8][9][10][11][12][13][14][15]).…”

Umbral Methods and Harmonic Numbers

“…Methods employing the concepts and the formalism of umbral calculus have been exploited in [1] to conjecture the existence of generating functions involving Harmonic Numbers [2]. The conjectures put forward in [1] have been proven in [3]- [4], further elaborated in subsequent papers [5] and generalized to Hyper-Harmonic Numbers in [6]. In this note we use the same point of view of [1] , by discussing the possibility of exploiting the formalism developed therein in a wider context.…”

“…with the property ĥn ĥm = ĥn+m (5) and introduce the Harmonic Based Exponential Function (HBEF) h e(x) = e ĥ x = 1…”