Expressing the semi-quantum operators, of Meixner random variables in terms of the position and momentum operators

Abstract:

The quantum and semi-quantum operators, generated by a probability measure having finite moments of all orders, are presented first. We continue by giving a general proposition about writing these operators as a sum of products (compositions) of multiplication by X and differentiation, D operators, in which all the X factors are to the left of the D factors. We show some concrete computations, of the semi-quantum operators, for the random variables that belong to the Meixner class. Finally, we show how these computations can be used to recover all the distributions from the Meixner class.