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.Comment: 6 page

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 Schr\"odinger 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 Sch\"odinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.Comment: 16 pages, 9 figure

An operational point of view to the theory of multi-variable/multi-index Hermite polynomials

The use of algebraic tools of operational and umbral nature is exploited to develop a new point of view and to extend the theory of Hermite polynomials, with more than one variable also of complex nature. The techniques we adopt includes multivariable/many index Hermite- Kampe-de-Feriet polynomials of order two and higher. It will be shown that the treatment, foreseen here, simplifies the study of the relevant properties and the associated computational technicalities

Space Charge and Quantum Corrections in Free Electron Laser Evolution

Effects producing gain dilution in Free Electron Laser devices are well documented. We develop here a unified point of view allowing the introduction of space charge effects, along with the gain deterioration due to inhomogeneous broadening contributions and discuss the relevant interplay. We outline future developments and comment on the possibility of including in the formalism effects of quantum mechanical nature.Comment: 10 pages, 7 figure

On an Umbral Point of View of the Gaussian and Gaussian-like Functions

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

On an Umbral point of view to the Gaussian and Gaussian like functions

In this note we review the theory of Gaussian functions by exploiting a point of view based on symbolic methods of umbral nature. We introduce quasi-Gaussian functions, which are close to Gaussian distribution but have a longer tail. Their use and their link with hypergeometric function is eventually presented

Mathematical Methods for Physicists

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Voigt Transform and Umbral Image

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