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Search Results (4)

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Keywords = umbral algebra

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2 pages, 180 KiB  
Editorial
Special Issue Editorial “Special Functions and Polynomials”
by Paolo Emilio Ricci
Symmetry 2022, 14(8), 1503; https://doi.org/10.3390/sym14081503 - 22 Jul 2022
Viewed by 1128
Abstract
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several [...] Read more.
This Special Issue contains 14 articles from the MDPI journal Symmetry on the general subject area of “Special Functions and Polynomials”, written by scholars belonging to different countries of the world. A similar number of submitted articles was not accepted for publication. Several successful Special Issues on the same or closely related topics have already appeared in MDPI’s Symmetry, Mathematics and Axioms journals, in particular those edited by illustrious colleagues such as Hari Mohan Srivastava, Charles F. Dunkl, Junesang Choi, Taekyun Kim, Gradimir Milovanović, and many others, who testify to the importance of this matter for its applications in every field of mathematical, physical, chemical, engineering and statistical sciences. The subjects treated in this Special Issue include, in particular, the following Keywords. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
18 pages, 302 KiB  
Article
Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study
by Giuseppe Dattoli, Silvia Licciardi and Rosa Maria Pidatella
Symmetry 2021, 13(5), 781; https://doi.org/10.3390/sym13050781 - 1 May 2021
Cited by 5 | Viewed by 2246
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Special Functions and Polynomials)
11 pages, 268 KiB  
Article
Dual Numbers and Operational Umbral Methods
by Nicolas Behr, Giuseppe Dattoli, Ambra Lattanzi and Silvia Licciardi
Axioms 2019, 8(3), 77; https://doi.org/10.3390/axioms8030077 - 2 Jul 2019
Cited by 7 | Viewed by 5467
Abstract
Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a [...] Read more.
Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus. Full article
(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)
12 pages, 800 KiB  
Article
Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods
by Yilmaz Simsek
Axioms 2018, 7(2), 22; https://doi.org/10.3390/axioms7020022 - 1 Apr 2018
Cited by 14 | Viewed by 4509
Abstract
In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order [...] Read more.
In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers y 1 n , k ; λ . Finally, we make some remarks and observations regarding these identities and relations. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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