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Symmetry
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Umbral Calculus, Operator Theory and Symmetry: Applications of Different...
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Umbral Calculus, Operator Theory and Symmetry: Applications of Different Mathematical Languages

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 3946

Special Issue Editors

Dr. Silvia Licciardi

E-Mail Website
Guest Editor
ENEA—Frascati Research Center, Via Enrico Fermi 45, 00044 Rome, Italy
Interests: umbral calculus; special functions; combinatorics; fractional dynamics; evolution and diffusion models; PDE; ODE; Volterra equations; operator theory; mathematical analysis; trigonometry
Prof. Dr. Giuseppe Dattoli

E-Mail Website
Guest Editor
ENEA/Fusion Department, Frascati Research Center, Rome, Italy
Interests: lasers; accelerators; free electron lasers; applied mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The properties of symmetry have been central to the conception of umbral calculus, which were originally developed from the calculus of differences and, more recently, on the extension of Weyl—Heisenberg group.

Umbral calculus deepens its roots into the Heaviside operational methods and into the techniques introduced since the 17th century, within the context of calculus of differences and by the English operationalist school of the 19th century. Since then, much work has been done to increase the rigor of and expand the original techniques to the extent of a much broader and more complete theory. In the second half of the last century, the umbral calculus of Roman and Rota, the theory of poweroids, the monomiality principle and the umbral indicial calculus provided further breakthroughs in this direction.

Umbral techniques make use of numerous transversal tools in various fields of mathematics, such as special functions, integral transforms, combinatorics, operator theory, mathematical analysis, and so on. The umbral formalism is a tool that, in addition to entering more deeply into the structure of the functions themselves, allows the creation of a link with other or new families of functions or polynomials, to treat ODEs and PDEs, fractional calculus, introduce different “trigonometries”, and much more.

In this perspective, this Special Issue aims to provide a broad perspective of how it is possible to treat different problems in pure and applied math by the use of a unifying formalism capable of entering more deeply into the mathematical structure of the formalism itself and of opening to pioneering methods.

Dr. Silvia Licciardi
Prof. Dr. Giuseppe Dattoli
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (4 papers)

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Research

13 pages, 461 KiB  
Article
On an Umbral Point of View of the Gaussian and Gaussian-like Functions
by Giuseppe Dattoli, Emanuele Di Palma and Silvia Licciardi
Symmetry 2023, 15(12), 2157; https://doi.org/10.3390/sym15122157 - 04 Dec 2023
Viewed by 776
Abstract
The theory of Gaussian functions is reformulated using an umbral point of view. The symbolic method we adopt here allows an interpretation of the Gaussian in terms of a Lorentzian image function. The formalism also suggests the introduction of a new point of [...] Read more.
The theory of Gaussian functions is reformulated using an umbral point of view. The symbolic method we adopt here allows an interpretation of the Gaussian in terms of a Lorentzian image function. The formalism also suggests the introduction of a new point of view of trigonometry, opening a new interpretation of the associated special functions. The Erfi(x), is, for example, interpreted as the “sine” of the Gaussian trigonometry. The possibilities offered by the Umbral restyling proposed here are noticeable and offered by the formalism itself. We mention the link between higher-order Gaussian trigonometric functions, Hermite polynomials, and the possibility of introducing new forms of distributions with longer tails than the ordinary Gaussians. The possibility of framing the theoretical content of the present article within a redefinition of the hypergeometric function is eventually discussed. Full article
(This article belongs to the Special Issue Umbral Calculus, Operator Theory and Symmetry: Applications of Different Mathematical Languages)
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13 pages, 1178 KiB  
Article
Fractional Differential Equations and Expansions in Fractional Powers
by Diego Caratelli, Pierpaolo Natalini and Paolo Emilio Ricci
Symmetry 2023, 15(10), 1842; https://doi.org/10.3390/sym15101842 - 29 Sep 2023
Cited by 2 | Viewed by 673
Abstract
We use power series with rational exponents to find exact solutions to initial value problems for fractional differential equations. Certain problems that have been previously studied in the literature can be solved in a closed form, and approximate solutions are derived by constructing [...] Read more.
We use power series with rational exponents to find exact solutions to initial value problems for fractional differential equations. Certain problems that have been previously studied in the literature can be solved in a closed form, and approximate solutions are derived by constructing recursions for the relevant expansion coefficients. Full article
(This article belongs to the Special Issue Umbral Calculus, Operator Theory and Symmetry: Applications of Different Mathematical Languages)
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22 pages, 591 KiB  
Article
Evaluation of Hamiltonians from Complex Symplectic Matrices
by Gianfranco Cariolaro and Alberto Vigato
Symmetry 2023, 15(5), 1000; https://doi.org/10.3390/sym15051000 - 28 Apr 2023
Viewed by 1184
Abstract
Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the [...] Read more.
Gaussian unitaries play a fundamental role in the field of continuous variables. In the general n mode, they may formulated by a second-order polynomial in the bosonic operators. Another important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link from symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non-trivial question: which Hamiltonian produces a given symplectic representation? The complex instead of the traditional real symplectic representation is considered, with the advantage of getting compact and elegant relations. The application to the single, two, and three modes illustrates the theory. Full article
(This article belongs to the Special Issue Umbral Calculus, Operator Theory and Symmetry: Applications of Different Mathematical Languages)
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10 pages, 296 KiB  
Article
Certain Properties of Δh Multi-Variate Hermite Polynomials
by Ibtehal Alazman, Badr Saad T. Alkahtani and Shahid Ahmad Wani
Symmetry 2023, 15(4), 839; https://doi.org/10.3390/sym15040839 - 31 Mar 2023
Cited by 1 | Viewed by 753
Abstract
The research described in this paper follows the hypothesis that the monomiality principle leads to novel results that are consistent with past knowledge. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite polynomials [...] Read more.
The research described in this paper follows the hypothesis that the monomiality principle leads to novel results that are consistent with past knowledge. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite polynomials ΔhHm(q1,q2,⋯,qr;h). We obtain their recurrence relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families, and thus the operational principles satisfied by these polynomials are derived and will prove beneficial for future observations. Full article
(This article belongs to the Special Issue Umbral Calculus, Operator Theory and Symmetry: Applications of Different Mathematical Languages)
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