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

A note on a Poissonian functional and a $q$-deformed Dufresne identity