Transformations of Laguerre 2D polynomials with applications to quasiprobabilities

Laguerre 2D polynomials are defined and their properties are investigated. The Laguerre 2D functions, introduced in [1, 2] are related to the Laguerre 2D polynomials in such a way that they also include the weight function for the orthonormalization of the Laguerre 2D polynomials. A one-parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied, in particular, to Hermite polynomials and to Laguerre 2D polynomials. These transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments. They contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for Hermite and Laguerre 2D polynomials. Applications of the Laguerre 2D polynomials and 2D functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the Fock-state basis and by normally and otherwise ordered moments are considered. The inversion of these representations is obtained in all cases. A restricted design of quasiprobabilities should become possible.