Random deletion model and Rainville's generating function for the Laguerre polynomials (Q5959118)

Here the author uses a random deletion model associated with the non-central negative binomial distribution to give a new proof of the following well-known generating function formula of Rainville for the generalized Laguerre polynomials NEWLINE\[NEWLINE\sum^\infty_{m=0} {m+n\choose n} z^mL^{(\nu-1)}_{m+n} (x) ={1\over(1-z)^{\nu +n}} L_n^{(\nu-1)} \left({x\over 1-z} \right)\exp \left({-xz \over 1-z}\right).NEWLINE\]NEWLINE The proof, using only probabilistic arguments, relies on the Poisson mixture representation of the distribution and the cumulative nature of the deletion and mixing operations.