Two differential equations for the linear generating function of the charlier polynomials

We provide an interesting way to obtain the linear generating function for the classical discrete Charlier orthogonal polynomials by implementing what we entitle the 'Inverse Method'. This method transforms a given three-term recurrence relation into a differential equation, the solution of which is a linear generating function. To demonstrate the details of the procedure, we first apply the Inverse Method to the three-term recurrence relation that defines the Charlier polynomials. We then apply it to a new three-term recurrence relation, which is established via a certain connection between the Charlier polynomials and a variation of the Laguerre polynomials. The solution to each of these differential equations is the intended generating function.