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27 pages, 341 KiB

Open AccessFeature PaperArticle

Symbolic Methods Applied to a Class of Identities Involving Appell Polynomials and Stirling Numbers

by Tian-Xiao He and Emanuele Munarini

Mathematics 2025, 13(11), 1732; https://doi.org/10.3390/math13111732 - 24 May 2025

Viewed by 296

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related [...] Read more.

In this paper, we present two symbolic methods, in particular, the method starting from the source identity, umbra identity, for constructing identities of s-Appell polynomials related to Stirling numbers and binomial coefficients. We discuss some properties of s-Appell polynomial sequences related to Riordan arrays, Sheffer matrices, and their q analogs. Full article

(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)

12 pages, 249 KiB

Open AccessArticle

Certain Summation and Operational Formulas Involving Gould–Hopper–Lambda Polynomials

by Maryam Salem Alatawi

Mathematics 2025, 13(2), 186; https://doi.org/10.3390/math13020186 - 8 Jan 2025

Viewed by 584

Abstract

This manuscript introduces the family of Gould–Hopper–Lambda polynomials and establishes their quasi-monomial properties through the umbral method. This approach serves as a powerful mechanism to analyze the characteristic of multi-variable special polynomials. Several summation formulas for these polynomials are explored, and their operational [...] Read more.

This manuscript introduces the family of Gould–Hopper–Lambda polynomials and establishes their quasi-monomial properties through the umbral method. This approach serves as a powerful mechanism to analyze the characteristic of multi-variable special polynomials. Several summation formulas for these polynomials are explored, and their operational identities are obtained using partial differential equations. The corresponding results for Hermite–Lambda polynomials are also obtained. In addition, a conclusion is given. Full article

18 pages, 274 KiB

Open AccessArticle

Solutions of Umbral Dirac-Type Equations

by Hongfen Yuan and Valery Karachik

Mathematics 2024, 12(2), 344; https://doi.org/10.3390/math12020344 - 20 Jan 2024

Viewed by 1366

Abstract

The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated [...] Read more.

The aim of this work is to study the method of the normalized systems of functions. The normalized systems of functions with respect to the Dirac operator in the umbral Clifford analysis are constructed. Furthermore, the solutions of umbral Dirac-type equations are investigated by the normalized systems. Full article

(This article belongs to the Special Issue Differential Equations with Boundary Value Problems: Theory and Applications)

13 pages, 461 KiB

Open AccessArticle

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 - 4 Dec 2023

Viewed by 1471

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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19 pages, 538 KiB

Open AccessArticle

Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family

by Tabinda Nahid and Junesang Choi

Fractal Fract. 2022, 6(4), 211; https://doi.org/10.3390/fractalfract6040211 - 9 Apr 2022

Cited by 9 | Viewed by 2186

Abstract

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials [...] Read more.

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

7 pages, 253 KiB

Open AccessArticle

q-Functions and Distributions, Operational and Umbral Methods

by Giuseppe Dattoli, Silvia Licciardi, Bruna Germano and Maria Renata Martinelli

Mathematics 2021, 9(21), 2664; https://doi.org/10.3390/math9212664 - 21 Oct 2021

Cited by 1 | Viewed by 1497

Abstract

The use of non-standard calculus means have been proven to be extremely powerful for studying old and new properties of special functions and polynomials. These methods have helped to frame either elementary and special functions within the same logical context. Methods of Umbral [...] Read more.

The use of non-standard calculus means have been proven to be extremely powerful for studying old and new properties of special functions and polynomials. These methods have helped to frame either elementary and special functions within the same logical context. Methods of Umbral and operational calculus have been embedded in a powerful and efficient analytical tool, which will be applied to the study of the properties of distributions such as Tsallis, Weibull and Student’s. We state that they can be viewed as standard Gaussian distributions and we take advantage of the relevant properties to infer those of the aforementioned distributions. Full article

(This article belongs to the Special Issue Advances in Functional Equations and Convex Analysis)

18 pages, 302 KiB

Open AccessArticle

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 4 | Viewed by 2225

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)

12 pages, 257 KiB

Open AccessArticle

Voigt Transform and Umbral Image

by Silvia Licciardi, Rosa Maria Pidatella, Marcello Artioli and Giuseppe Dattoli

Math. Comput. Appl. 2020, 25(3), 49; https://doi.org/10.3390/mca25030049 - 31 Jul 2020

Cited by 2 | Viewed by 2172

Abstract

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other [...] Read more.

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions. Full article

11 pages, 268 KiB

Open AccessEditor’s ChoiceArticle

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 5440

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, 300 KiB

Open AccessArticle

Some Properties and Generating Functions of Generalized Harmonic Numbers

by Giuseppe Dattoli, Silvia Licciardi, Elio Sabia and Hari M. Srivastava

Mathematics 2019, 7(7), 577; https://doi.org/10.3390/math7070577 - 28 Jun 2019

Cited by 13 | Viewed by 3930

Abstract

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as [...] Read more.

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)

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16 pages, 346 KiB

Open AccessArticle

Operator Ordering and Solution of Pseudo-Evolutionary Equations

by Nicolas Behr, Giuseppe Dattoli and Ambra Lattanzi

Axioms 2019, 8(1), 35; https://doi.org/10.3390/axioms8010035 - 16 Mar 2019

Cited by 1 | Viewed by 3436

Abstract

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and [...] Read more.

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance. Full article

(This article belongs to the Special Issue Fractional Differential Equations)

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10 pages, 251 KiB

Open AccessArticle

Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

by Taekyun Kim and Dae San Kim

Mathematics 2019, 7(1), 47; https://doi.org/10.3390/math7010047 - 4 Jan 2019

Cited by 6 | Viewed by 2641

Abstract

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate [...] Read more.

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials. Full article

(This article belongs to the Special Issue Polynomials: Theory and Applications)

9 pages, 243 KiB

Open AccessArticle

Umbral Methods and Harmonic Numbers

by Giuseppe Dattoli, Bruna Germano, Silvia Licciardi and Maria Renata Martinelli

Axioms 2018, 7(3), 62; https://doi.org/10.3390/axioms7030062 - 1 Sep 2018

Cited by 7 | Viewed by 3786

Abstract

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications)

12 pages, 800 KiB

Open AccessArticle

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 4496

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)

11 pages, 289 KiB

Open AccessArticle

From Circular to Bessel Functions: A Transition through the Umbral Method

by Giuseppe Dattoli, Emanuele Di Palma, Silvia Licciardi and Elio Sabia

Fractal Fract. 2017, 1(1), 9; https://doi.org/10.3390/fractalfract1010009 - 8 Nov 2017

Cited by 13 | Viewed by 3458

Abstract

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family [...] Read more.

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed. Full article

(This article belongs to the Special Issue Fractional Dynamics)

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