Voigt Transform and Umbral Image

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 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.

Umbral Calculus and New Trigonometries

“…The same consideration can be done for each special function through appropriate umbral images. We list in the following just some examples for the Laguerre, Jacobi, Legendre, Tricomi-Bessel and Chebyshev functions, and the Voigt transform [12,17,18].…”

Integrals of Special Functions and Umbral Methods

“…We will exploit the above umbral restyling by deriving an integral involving Gaussian and Lorentzian functions. 2 We use the Laplace transform identity…”

“…Methods of algebraic and umbral nature possess, among the other advantages, the undoubtful merit of simplifying practical computational issues. In a numbers of previous papers [1][2][3][4][5][6][7] it has been established that the one of the possible umbral images of a Bessel function is a Gaussian. This statement can be profitably exploited in the present context, provided we establish some reference tools [6,8].…”

“…Definition 2 We introduce the operatorĉ called "umbral", c := e ∂ z , (2) as the vacuum shift operator, with z the domain's variable of the function on which the operator acts.…”