arXiv subjects
Publications and source records attributed to G. Dattoli.
We show that the use of generalized multivariable forms of Hermite polynomials provide an useful tool for the evaluation of families of elliptic type integrals often encountered in electrostatic and electrodynamics
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid polynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation $[\hat P,\hat M]=1$. In ordinary quantum mechanics $\hat P$ is the derivative and $\hat M$ the coordinate operator. Here we shall realize $\hat P$ as a second order differential operator and $\hat M$ as a first order integral one. We show that this makes it possible to solve large classes of differential and integro-differential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing so called flatenned beams in laser theory
We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the general properties of these transformations, derive explicit examples and discuss them in the case of the Appèl and Sheffer polynomial families. We apply these transformations to non-linear equations, and discuss how the relevant solutions should be interpreted.
The study presented here was initiated by a discussion to investigate the possibility of using synchrotron radiation as a source for the Next Generation Lithography (NGL) based on the EUV-concept (Extreme Ultra-Violet; here 13.5 nm or 11.3 nm radiation, respectively). The requirements are: 50 W, 2% bandwidth and minimal power outside this bandwidth. Three options were investigated. The first two deal with radiation from bending magnets and undulators. The results confirm the earlier work by Oxfords Instrument and others that these light-sources lack in-band power while emitting excessive out-of-band radiation. The third approach is a FEL (Free Electron Laser) driven by a 500 MeV linear accelerator with a superconducting mini-undulator as radiation emitting device. Such a device would produce in-band EUV-power in excess of 50 W with negligible out-of-band power.