Random deletion model and Rainvilleʹs generating function for the laguerre polynomials

A random deletion model associated with the non-central negative binomial distribution is used to provide a new proof of the well-known generating function formula of Rainville for the generalised Laguerre polynomials. The proof, using only probabilistic arguments, relies on the Poisson mixture representation of the distribution and the commutative nature of the deletion and mixing operations.