  In this note, we compute the Mellin transform of a Poissonian exponential
functional, the underlying process being a simple continuous time random walk.
It shows that the Poissonian functional can be expressed in term of the inverse
of a $q$-gamma random variable.
  The result interpolates between two known results. When the random walk has
only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor.
In the Brownian limit ($q \rightarrow 1^-$), one recovers Dufresne's identity
involving an inverse gamma random variable. Hence, one can see it as a
$q$-deformed Dufresne identity.
