The Airy transform and associated polynomials

Danilo Babusci 1 , Giuseppe Dattoli 2 , and Dario Sacchetti 3
  • 1 Laboratori Nazionali di Frascati, INFN, via E. Fermi 40, I-00044, Frascati, Italy
  • 2 Centro Ricerche Frascati, ENEA, via E. Fermi 45, I-00044, Frascati, Italy
  • 3 Dipartimento di Statistica Probabilità e Statistica Applicata, Università “Sapienza”, P.le A. Moro, 5, 00185, Roma, Italy

Abstract

The Airy transform is an ideally suited tool to treat problems in classical and quantum optics. Even though the relevant mathematical aspects have been thoroughly investigated, the possibilities it offers are wide and some features, such as the link with special functions and polynomials, still contain unexplored aspects. In this note we will show that the so called Airy polynomials are essentially the third order Hermite polynomials. We will also prove that this identification opens the possibility of developing new conjectures on the properties of this family of polynomials.

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