Laguerre polynomials: novel properties and numerical generation scheme for analysis of a discrete probability distribution

We compute a quantum mechanical probability distribution expressed as a series of generalized Laguerre polynomials. A method for the numerical generation of the polynomials of any integer order is given. Two new properties of these polynomials are derived to reduce the distribution to a closed form expression involving only a single polynomial and to devise a criterion for studying the accuracy of the generation procedure. (orig.)