Let 
be any function that factors as 
. For such functions, the Z transform 
is useful in simplifying calculations involving sums 
, where the summation extends over all 
having non-negative integer components and 
. Define the discrete convolution product of two functions g and h by 
. (Note that 
is both commutative and associative, so that the order that the convolutions are taken in is irrelevant, justifying the use of the above notation when several functions are involved.)
The Z transform convolution theorem may be thought of as a discretized form of the Laplace convolution theorem (see theorem 2).
: If 
where the function 
then 
and
, for all z such that 
, 
, converges.
Proof: For m=2 we have 
and the Z transforms of 
and 
are given by 
, i=1,2, respectively. For z within the radii of convergence of both of these power series, we have (after collecting terms having the same power of z)
. The right-hand side is immediately seen to be Z[F](z). The result for arbitrary m follows by induction. QED.
Note that due to the uniqueness of power series representations, inverses of Z transforms exist on the nonnegative integers.