A. Torre

Theory of Generalized Bessel Functions. - II.

SummaryIn this paper we continue the systematic study of the generalized Bessel functions (GBF) recently introduced and often encountered in problems of scattering for which the dipole approximation is inadequate. We analyse the relations among different GBF and discuss their importance for the solution of differential finite-difference equation of the Raman-Nath type. We present numerical results for the first-kind cylinder GBF in the preasymptotic region and also a preliminary analysis of the asymptotic properties of the modified GBF.

Generalized polynomials and associated operational identities

We use operational identities to introduce multivariable Laguerre polynomials. We explore the wealth of differential equations they satisfy. We analyze their properties and the link with Legendre-type polynomials.

European project to develop a UV/VUV free-electron laser facility on the ELETTRA storage ring☆

Abstract The main features and novel technical aspects of a new European project to integrate a free-electron laser on an existing “third generation” synchrotron radiation user facility are described, including the design of the optical cavity and undulator, the electron beam characteristics and a first assessment of the predicted laser performance.

Design considerations on a high-power VUV FEL

We explore the feasibility conditions of a high-power EEL operating in the VUV region (below 100 nm) and exploiting a coupled oscillator triplicator configuration. A high quality beam from a linac is passed through a EEL oscillator and produces laser radiation at 240 nm. The same beam is extracted and then injected into a second undulator tuned at the third harmonic of the first. The bunching produced in the oscillator allows the start up of the laser signal in the second section which operates as an amplifier. We discuss the dynamical behavior of the system and the dependence of the output power on the characteristics of the e-beam and of the oscillator. The possibility of enhancing the output power, adding a tapered section to the second undulator, is finally analyzed. >

Generating functions of multivariable generalized Bessel functions and Jacobi‐elliptic functions

It is pointed out that the Jacobi‐elliptic functions are the natural basis to get generating functions of the multivariable generalized Bessel functions. Analytical and numerical results are given of interest for applications.

Linear and radial canonical transforms of fractional order

The Laplace and Barut-Girardello transforms of fractional order are introduced and their relations to canonical transformations, parabolic differential equations and special functions are discussed.

Exponential operators, quasi-monomials and generalized polynomials

Abstract It is shown that the combination of exponential operator techniques and the use of the principle of quasimonomiality can be a very useful tool for a more general insight into the theory of ordinary polynomials and for their extension. We prove the link between Laguerre polynomials and Tricomi functions and study the properties of new classes of polynomials constructed in terms of quasi-monomials. The usefulness of the obtained results to treat radiation physics problems such as wave propagation and quantum beam life-time in storage rings is also discussed.

Theory of two‐index Bessel functions and applications to physical problems

In this article the theory of two‐index Bessel functions is presented. Their generating function, series expansion, and integral representations are discussed. Their usefulness in physical problems is also discussed in the context of analysis of radiation emitted by relativistic electrons in two‐frequency undulators. Finally, the theoretical analysis proving addition and multiplication theorems for two‐index Bessel functions are completed and their modified forms are introduced.

Phase‐space dynamics and Hermite polynomials of two variables and two indices

The theory of Hermite polynomials of two variables and two indices is discussed herein. Within the context of phase‐space formulation of classical and quantum mechanics, they play the same role as conventional Hermite polynomials in ordinary quantum mechanics. Finally their extension to m variables and m indices is analyzed.

The generalized Laguerre polynomials, the associated Bessel functions and application to propagation problems

Abstract The theory of generalized Bessel functions has found significant applications in the analysis of radiation phenomena, associated with charges moving in magnetic devices. In this paper we exploit the monomiality principle to discuss the theory of two-variable Laguerre polynomials and introduce the associated Laguerre–Bessel functions. We study their properties (addition and multiplication theorems, generating function, recurrence relations and so on) and discuss the link with the ordinary case. The usefulness of the obtained results to treat problems relevant to the paraxial propagation of electromagnetic waves is also discussed.