Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

Giuseppe Dattoli,Silvia Licciardi,Rosa Maria Pidatella

doi:10.3390/sym13050781

Giuseppe Dattoli, Silvia Licciardi + Show 1 more

Open Access

https://doi.org/10.3390/sym13050781

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Abstract

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

Highlights

IntroductionThe Authors of this paper have largely benefitted from the techniques suggested by umbral methods and have embedded them with other means associated, e.g., with algebraic operational procedure, to obtain new results and to reformulate previous, apparently extraneous topics, within a unifying point of view
New concepts and techniques emerged in the past within the framework of special functions have had positive feedback in other and more abstract fields of Mathematics
The techniques associated with umbral calculus have opened new and unexpected avenues in analysis and simplified the technicalities of calculations, which are awkwardly tedious when performed with conventional computational means

Summary

Introduction

The Authors of this paper have largely benefitted from the techniques suggested by umbral methods and have embedded them with other means associated, e.g., with algebraic operational procedure, to obtain new results and to reformulate previous, apparently extraneous topics, within a unifying point of view. We embed umbral methods and formal integration techniques to explore the field of number series by getting a non-conventional treatment of problems usually treated with completely different means. The underlying formalism allows the handling of integrals and derivatives on the same footing Within this framework the primitive of the product of two functions is nothing but a restatement of the Leibniz formula [3,4] as it has been proved in ref. The previous results are quite surprising, they are associated with generalized forms of telescopic series and can be embedded in an even wider context, as proved in the forthcoming section

Combinatorial Identities and Leibniz Formula

Inverse Derivatives and Generalized Telescopic Series

Umbral Methods and Binomial Harmonic Numbers

Combinatorial Identities and Integral Representations

Final Comments

Further Results