Heisenberg algebra, umbral calculus and orthogonal polynomials

Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [(P) over cap,(M) over cap]=1. In ordinary quantum mechanics, (P) over cap is the derivative and (M) over cap the coordinate operator. Here, we shall realize (P) over cap as a second order differential operator and (P) over cap as a first order integral one. We show that this makes it possible to solve large classes of differential and integrodifferential equations and to introduce new classes of orthogonal polynomials, related to Laguerre polynomials. These polynomials are particularly well suited for describing the so-called flatenned beams in laser theory (C) 2008 American Institute of Physics.