's domain
To apply the T transform, the assumption that all 
had to be made. Here we present a simple theorem that expands the region of validity of the various expressions derived in this paper to the region where any of the 
may be non-negative. We present the theorem for the single subset sum case only, although the multiple non-overlapping subset case and the contained overlap case may be handled in an almost identical manner.
: If 
, 
and 
, 
then
Proof: Note that 
implies that there is an integer q>0 and an 
such that 
. Thus 
may be rewritten as
where 
, the Kronecker delta function. Iterate this operation q times (removing one power from 
and summing with an increased count vector each time) to find
Simplify this to yield
where the vector 
has nonnegative integer components summing to q with 
for 
. Since , evaluate the integral 
using theorem 12 with k=1 (noting that 
and 
increase by q due to being added to 
) to find
Now, we put 
into closed form by noting that it is the discrete convolution product of the functions of 
of 
given by
Apply the Z transform convolution theorem (see appendix 9.6.2) to find
Note that
for 
and substitute for the Z transforms to find
Substituting this result in (*) and simplifying leads to the desired result. QED.
We resort to analytic continuation in the non-contained overlap case.