A variety of applied probability problems on words can be reduced to the study of the asymptotic behavior of the so called Poisson transform L (z, u) (where z and u are complex) which satisfies the following differential-functional equation
with, say 
, where 
, 
are constants b is a positive
integer, p + q = 1, and a (z,u) is a given function. This
equation arises, for example, in the following context: An integer
valued random variable 
is transformed into the probability
generating function 
as an
intermediate step in obtaining the Poisson transform 
that satisfies the
above equation. The Poisson version of the problem replaces the
deterministic input n by a Poisson variable N with mean z =
n. Such a poissonization is often useful since it leads to a
simpler solution (due to some unique properties of the Poisson
process). In this research we aim at providing an asymptotic
solution of the above (and others) differential-functional equation
for 
in a cone around real axis. This leads to a
solution of the problem at hand within the Poisson model framework.
To translate it into the original model one needs (subtle)
depoissonization results that are our second topic of the proposed
research. Most of our depoissonization findings fall into the
following general scheme: if the Poisson transform is
appropriately bounded in a cone around the real axis and does not
grow too fast outside such a cone, then one can asymptotically
depoissonize by replacing the Poisson process by its mean. Not
unexpectedly, actual formulations of depoissonization depend on the
nature of the bound, and thus we shall investigate diagonal
depoissonization theorems. Finally, we apply our results to a
variety of challenging problems on words arising in pattern
matching, data compression, security, cryptology, coding, molecular
biology, and so forth.