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G. Dattoli

Publications and source records attributed to G. Dattoli.

At least 19 recordsLinked to original sources
arXiv2024

Nearly cosine series and generalized trigonometric functions

A class of overlooked trigonometric-like functions is explored in this article, along with the relevant applications in applications. We show indeed that Taylor series, resembling that of an ordinary cosine, are representative of wider classes of functions, naturally suited for prolems ranging from molecular to Laser Physics. The article goes through the original motivations of the proposal and studies the relevant properties within the context of an Umbral interpretation. Their use in applications is discussed within the framework of Free Electron Laser theory, Lennard- Jones potentials and Kramers-Kronig causality identities

math-ph↗
arXiv2023

A note on differential equations of logistic type

Logistic equations play a pivotal role in the study of any non linear evolution process exhibiting growth and saturation. The interest for the phenomenology, they rule, goes well beyond physical processes and cover many aspects of ecology, population growth, economy...According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated non linear differential equations, useful to guess "hidden" evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing non standard forms of derivative operators, like those of the Laguerre type.

math.CA↗
arXiv2020

Generating Functions for Lacunary Legendre and Legendre-like Polynomials

The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new properties regarding the generating functions, repeated derivatives of irrational functions and so on. We finally extend the method of the integral transform to functions with more than one variable and derive new expansion criteria of a given function in terms of Legendre polynomials.

math.CA↗
arXiv2020

Umbral-Algebraic Methods and Asymptotic Properties of Special Polynomials

A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial families, with the use of elementary algebraic tools.

math.CA↗
arXiv2019

On the Sheffer-type polynomials related to the Mittag-Leffler functions: applications to fractional evolution equations

We present two types of polynomials related to the Mittag-Leffler function namely the fractional Hermite polynomial and the Mittag-Leffler polynomial. The first modifies the Hermite polynomial and the second one is a refashioned Laguerre polynomial. The fractional Hermite and the Mittag-Leffler polynomials are used to solve {the Cauchy problems for} the fractional Fokker-Planck equation where the fractional derivative is taken in the Caputo sense with respect to time and/or space. The generating functions of these two kinds of polynomials are also calculated and they indicate that these polynomials belong to the Sheffer type.

math-ph↗
arXiv2019

Can Umbral and $q$-calculus be merged?

The $q$-calculus is reformulated in terms of the umbral calculus and of the associated operational formalism. We show that new and interesting elements emerge from such a restyling. The proposed technique is applied to a different formulations of $q$ special functions, to the derivation of integrals involving ordinary and $q$-functions and to the study of $q$-special functions and polynomials.

math.CA↗
arXiv2019

Operational versus umbral methods and the Borel transform

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

math.CA↗
arXiv2018

EuPRAXIA@SPARC_LAB Design study towards a compact FEL facility at LNF

On the wake of the results obtained so far at the SPARC\_LAB test-facility at the Laboratori Nazionali di Frascati (Italy), we are currently investigating the possibility to design and build a new multi-disciplinary user-facility, equipped with a soft X-ray Free Electron Laser (FEL) driven by a $\sim$1 GeV high brightness linac based on plasma accelerator modules. This design study is performed in synergy with the EuPRAXIA design study. In this paper we report about the recent progresses in the on going design study of the new facility.

physics.acc-ph↗
arXiv2017

Mittag-Leffler function and fractional differential equations

We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $α$ ($0 < α< 1$) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.

math-ph↗
arXiv2017

Lacunary Generating Functions for the Laguerre Polynomials

Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Laguerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.

math-ph↗
arXiv2017

The Havriliak-Negami relaxation and its relatives: the response, relaxation and probability density functions

We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain $\sim[1+(iωτ_{0})^α]^{-β}$ with $τ_{0}$ being some characteristic time. For $α= l/k< 1$ ($l$ and $k$ positive integers) and $β> 0$ we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization $β= (2-q)/(q-1)$ and $τ_{0} = (q-1)^{1/α}$ $(1 < q < 2)$ we show that for $0 < α< 1$ the response functions $f_{α, β}(t/τ_{0})$ go to the one-sided Lévy stable distributions when $q$ tends to one. Moreover, applying the self-similarity property of the probability densities $g_{α, β}(u)$, we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

cond-mat.stat-mech↗
arXiv2017

The stretched exponential behavior and its underlying dynamics. The phenomenological approach

We show that the anomalous diffusion equations with a fractional derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution operator is presented as integral transforms whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and different initial conditions.

cond-mat.stat-mech↗
arXiv2016

Theory of relativistic heat polynomials and one-sided Lévy distributions

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.

math-ph↗
arXiv2016

Relativistic Heat Equation via Lévy stable distributions: Exact Solutions

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case.

math-ph↗
arXiv2016

Hermite Calculus

We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of practical rules allowing significant simplifications in computational problems.

math.CA↗
arXiv2016

Deep Saturated Free Electron Laser Oscillators and Frozen Spikes

We analyze the behavior of Free Electron Laser (FEL) oscillators operating in the deep saturated regime and point out the formation of sub-peaks of the optical pulse. They are very stable configurations, having a width corresponding to a coherence length. We speculate on the physical mechanisms underlying their growth and attempt an identification with FEL mode locked structures associated with Super Modes. Their impact on the intra-cavity nonlinear harmonic generation is also discussed along with the possibility of exploiting them as cavity out-coupler.

physics.acc-ph↗