Paper Info
Title
Inverse Derivative Operator And Umbral Methods For The Harmonic Numbers And Telescopic Series Study
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.
Year
DOI
Venue
2021
10.3390/sym13050781
SYMMETRY-BASEL
Keywords
DocType
Volume
umbral methods 05A40, operators theory 44A99, 47B99, 47A62, special functions 33C52, 33C65, 33C99, 33B10, 33B15, integral calculus 97I50, harmonic numbers 05A99, 11B75, combinatorics 05A10, 11B75, gamma function 33B15, telescopic series 11B65, 11B75, 05A10
Journal
13
Issue
Citations 
PageRank 
5
0
0.34
References 
Authors
0
3
Name
Order
Citations
PageRank
Giuseppe Dattoli100.68
Silvia Licciardi202.03
Rosa Maria Pidatella300.68