Some Mathematical Properties for Marginal Model of Poisson-Gamma Distribution

Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We first obtain explicit representations for these polynomials in form of the Laguerre polynomials and the confluent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, differential equation, Rodrigueś formula, finite sums and integral transforms.