Fractional Poisson Analysis in Dimension one

Abstract

In this paper, we use a biorthogonal approach (Appell system) to construct and characterize the spaces of test and generalized functions associated to the fractional Poisson measure π λ,β , that is, a probability measure in the set of natural (or real) numbers. The Hilbert space L 2 (π λ,β ) of complex-valued functions plays a central role in the construction, namely, the test function spaces. The characterization of these spaces is realized via integral transforms and chain of spaces of entire functions of different types and order of growth. Wick calculus extends in a straightforward manner from Gaussian analysis to the present non-Gaussian framework. Finally, in Appendix B we give an explicit relation between (generalized) Appell polynomials and Bell polynomials.