Two parameterized deformed Poisson type operator and the combinatorial moment formula

In this paper, we shall introduce two parameterized deformation of the classical Poisson random variable from the viewpoint of noncommutative probability, namely $(q,s)$-Poisson type operator (random variable) on the two parameterized deformed Fock space, namely, the $(q,s)$-Fock space constructed by the weighted q-deformation approach in [11], [4] (see also [6]). The recurrence formula for the orthogonal polynomials of the $(q,s)$-deformed Poisson distribution is determined. Moreover we shall also give the combinatorial moment formula of the $(q,s)$-Poisson type operator by using the set partitions and the card arrangement technique with their statistics. Our method presented in this paper provides nice combinatorial interpretations to parameters, q and s. The deformation presented in this paper can be regarded as a generalization of the Al-Salam-Carlitz type, because the restricted case $s=q$ recovers the q-Charlier polynomials of Al-Salam-Carlitz type appeared in combinatorics [17].