Bispectrality of Charlier type polynomials

Given a finite set of positive integers G and polynomials , , with degree of equal to g, we associate to them a sequence of Charlier type polynomials defined from the Charlier polynomials by using certain Casoratian determinants (whose entries are the polynomials ). Charlier type polynomials are eigenfunctions of higher order difference operators. When , the polynomials are also orthogonal, and then satisfy a three term recurrente relation. In this paper, we prove that whatever the polynomials 's are, the Charlier type polynomials always satisfy higher order recurrence relations. We also introduce and characterize the algebra of difference operators associated to the recurrence relations satisfied by the sequence . Our characterization is constructive and surprisingly simple. As a consequence, we prove that is, essentially, the unique choice such that the polynomials are orthogonal with respect to a measure.