  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 $\pi_{\lambda,\beta}$, that is, a probability
measure in the set of natural (or real) numbers. The Hilbert space
$L^{2}(\pi_{\lambda,\beta})$ of complex-valued functions plays a central role
in the construction, namely, the test function spaces
$(N)_{\pi_{\lambda,\beta}}^{\kappa}$, $\kappa\in[0,1]$ is densely embedded in
$L^{2}(\pi_{\lambda,\beta})$. Moreover, $L^{2}(\pi_{\lambda,\beta})$ is also
dense in the dual
$((N)_{\pi_{\lambda,\beta}}^{\kappa})'=(N)_{\pi_{\lambda,\beta}}^{-\kappa}$.
Hence, we obtain a chain of densely embeddings
$(N)_{\pi_{\lambda,\beta}}^{\kappa}\subset
L^{2}(\pi_{\lambda,\beta})\subset(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. 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.
