Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation . In ordinary quantum mechanics, is the derivative and the coordinate operator. Here, we shall realize as a second order differential operator and 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
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