The multivariate Charlier polynomials as matrix elements of the Euclidean group representation on oscillator states

Abstract: A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These polynomials in d discrete variables are orthogonal on the product of d Poisson distributions. The accent is put on the d=2 case and the group theoretical setting is used to obtain the main properties of the polynomials: orthogonality and recurrence relations, difference equation, raising/lowering relations, generating function, hypergeometric and integral representations and explicit expression in terms of standard Charlier and Krawtchouk polynomials. The approach is seen to extend straightforwardly to an arbitrary number of variables. The contraction of SO(3) to E(2) is used to show that the bivariate Charlier polynomials correspond to a limit of the bivariate Krawtchouk polynomials.