In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure [Formula: see text], [Formula: see text], on the dual of Schwartz test function space [Formula: see text]. The Hilbert space [Formula: see text] of complex-valued functions is described in terms of a system of generalized Appell polynomials [Formula: see text] associated to the measure [Formula: see text]. The kernels [Formula: see text], [Formula: see text], of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system [Formula: see text], there is a generalized dual Appell system [Formula: see text] that is biorthogonal to [Formula: see text]. The test and generalized function spaces associated to the measure [Formula: see text] are completely characterized using an integral transform as entire functions.

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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.