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.
Highlights
We begin the introductory section by providing an overview of the special functions and their umbral images.One of the appealing features of the umbral (U) [1] treatment of special functions is that it provides a means to reduce the study of the relevant properties to that of the elementary families, by establishing an appropriate correspondence between the higher transcendental under study and a corresponding elementary function, usually indicated as the associated umbral image (U I) [2].We note that the Gaussian is the U I of the cylindrical Bessel functions and that the Newton binomial realizes the U I of Hermite polynomials
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
Hermite and Laguerre calculus, as we presented here, is a by-product of the umbral formalism
Summary
We begin the introductory section by providing an overview of the special functions and their umbral images. The relevant NOH-function integral representation can be written by the use of the Laplace transform identity according to the following proposition. Equation (23) is interesting since, as will be shown, it can be interpreted in terms of two variable generalized Bessel functions Before closing this introductory section, we discuss a further example, based on the definition of a family of polynomials belonging to the so-called hybrid family, because they share properties in between those of Hermite and Laguerre polynomials [8]. We noted that Hermite functions of negative order can be defined by means of infinite integrals yielding the relevant integral representation; the use of the formalism we are proposing may be useful in a wider context as, e.g., for the evaluation of definite integrals, as shown below. The computation of higher order terms in g0 becomes heavily complicated to accomplish with the methods described here, notwithstanding the equality between the present treatment and the results from Mathematica is a further confirmation of the reliability of the procedure
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