Aurel Stan
The Ohio State University at Marion, Marion, USA

If a random variable has finite moments of all orders, then its moments can be recovered from the number operator. Any function of the number operator can be written as a series, in which each term is a composition of a multiplication operator by a polynomial and a power of the differentiation operator. This series is called the position-momentum decomposition of the operator. We present first a general discussion of the random variables for which this series has only a finite number of non-zero terms, that means the function of the number operator belongs to the Weyl algebra. We then focus our attention to the case when this function is a polynomial of degree at most two and the position-momentum decomposition is at most quadratic in the differentiation operator, recovering the random variables whose orthogonal polynomials are the Hermite, Laguerre, and Jacobi polynomials.