## Abstract

Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials (GHP), which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the GHP, which are covariant under O(N) transformations. The GHP and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) ∼ su(1, 1) algebra.

Original language | English |
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Pages (from-to) | 427-448 |

Number of pages | 22 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 38 |

Issue number | 2 |

DOIs | |

State | Published - 14 Jan 2005 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy (all)