Markov processes and a multiple generating function of product of generalized Laguerre polynomials

From the spectral representation of the transition probability of birth-and-death processes, Karlin and McGregor show that the transition probability for the infinite server Markovian queue is in the form of a diagonal sum involving a product of Charlier polynomials. By using Meixner's bilinear generating formula for the Charlier polynomials and the Markov property, a multiple generating far the Charlier polynomials is deduced from the Chapman-Kolmogorov equation. The resulting formula possesses the same genre of a multiple generating function for the generalized Laguerre polynomials discussed by Messina and Paladimo, the explicit solution of which is recently given by the present author.