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].
Read full abstract