arXiv subjects

Giuseppe Dattoli

Publications and source records attributed to Giuseppe Dattoli.

29 records · Page 2Linked to original sources

arXiv2019

Repeated derivatives of tanh, sech, ... and associated polynomials

Elementary problems like the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated by the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec(.), tan(.) and for their hyperbolic counterparts.

math.CA↗

arXiv2019

On the role of Hermite-like polynomials in the Fock representations of Gaussian states

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.

quant-ph↗

arXiv2019

Dual Numbers and Operational Umbral Methods

Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

math.GM↗

arXiv2019

Operator Ordering and Solution of Pseudo-Evolutionary Equations

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 of fractional differential equations. We apply the method to a number of examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

math-ph↗

arXiv2019

Operational Methods in the Study of Sobolev-Jacobi Polynomials

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of SJ polynomials explicitly.

math-ph↗

arXiv2017

Comments on the Properties of Mittag-Leffler Function

The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schrödinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

math-ph↗

arXiv2017

Umbral Methods and Harmonic Numbers

The theory of harmonic based function 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.

math.CA↗

arXiv2017

Motzkin Numbers: an Operational Point of View

The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory of special functions and with the relevant treatment in terms of operational means. The use of umbral methods opens new directions for further developments and generalizations, which leads, e.g., to the identification of new Motzkin associated forms.

math.CO↗

arXiv2017

Theory of Generalized Trigonometric Functions: from Laguerre to Airy Forms

We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired to the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transform allowing the framing of the previous concepts within the context of generalized Borel tranforms.

math.CA↗

arXiv2010

The fine structure constant and numerical alchemy

We comment on past and more recent efforts to derive a formula yielding the fine structure constant in terms of integers and transcendent numbers. We analyse these "exoteric" attitudes and describe the myths regarding α, which seems to have very ancient roots, tracing back to Cabbala and to medieval alchemic conceptions. We discuss the obsession for this constant developed by Pauli and the cultural "environment" in which such an "obsession" grew. We also derive a simple formula for α in terms of two numbers π and 137 only. The formula we propose reproduces the experimental values up to the last significant digit, it has not any physical motivation and is the result of an alchemic combination of numbers. We make a comparison with other existing formulae, discuss the relevant limits of validity by comparison with the experimental values and discuss a criterion to recover a physical meaning, if existing, from their mathematical properties.

physics.gen-ph↗

arXiv2010

Hyperbolic and Circular Trigonometry and Application to Special Relativity

We discuss the most elementary properties of the hyperbolic trigonometry and show how they can be exploited to get a simple, albeit interesting, geometrical interpretation of the special relativity. It yields indeed a straightforword understanding of the Lorentz transformation and of the relativistic kinematics as well. The geometrical framework adopted in the article is useful to disclose a wealth of alternative trigonometries not taught in undergraduate and graduate courses. Their introduction could provide an interesting and useful conceptual tool for students and teachers.

math-ph↗