The standard Charlier polynomials \(C_n(x;a)\) of degree \(n\) in the variable \(x\) were introduced in 1905, and can be defined through their generating function \[ e^t\Biggl(1-{t\over a}\Biggr)^x= \sum^\infty_{n=0} C_n(x; a)\,t^n/n!. \] These polynomials appear in various field, including combinatorics, statistics and probability; or physics.
In the last domain, the importance of them is due mostly to their appearance in the matrix elements of unitary irreducible representations of the one-dimensional oscillatory group [see e.g. Y. A. Granorskii and A. Zhedanov, Phys. J. 29, 387–393 (1986); or N. Ja. Vilenkin and A. U. Klimyk, Mathematics and Its Applications. Soviet Series. 72. Dordrecht etc.: Kluwer Academic Publishers. xxiii, 608 p. (1991; Zbl 0742.22001)].
In the paper under review, a new family of multivariable Charlier type polynomials that arise as matrix elements of the unitary reducible Euclidean group representation on oscillatory states is introduced. The main properties of these polynomials are studied.