Heisenberg algebra, umbral calculus and orthogonal polynomials

@article{Dattoli2008HeisenbergAU,
  title={Heisenberg algebra, umbral calculus and orthogonal polynomials},
  author={Giuseppe Dattoli and Decio Levi and Pavel Winternitz},
  journal={Journal of Mathematical Physics},
  year={2008},
  volume={49},
  pages={053509}
}
Umbral calculus can be viewed as an abstract theory of the Heisenberg commutation relation [P,M]=1. In ordinary quantum mechanics, P is the derivative and M the coordinate operator. Here, we shall realize P as a second order differential operator and M 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… Expand

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