. An alternate proof of theorem 12 leads to an identity. Where the inverse T transform is applied in the proof of theorem 12, instead find the convolution 
using theorem 10.2 and express it in terms of 
. Now, do the inverse transform and equate this result to the result of theorem 12 to find Gauss's identity:
. An alternate proof of theorem 14a leads to another identity. Instead of applying theorem 11, apply theorem 10.1 to the first two terms of the pairwise overlap convolution 
and immediately take the inverse 
transform. Now, do the final convolution with 
and take the inverse 
transform. Note that there is only a single summation in the result, whereas in the result in theorem 14a there are two summations. On the other hand, the convolution 
can be taken first, followed by a convolution with 
, effectively interchanging indices 1 and 2. Equating these two single-sum forms gives the identity
while equating either of the single-sum results just described to the original result of theorem 14a yields Gauss's identity equation 9.120, above. (To make the symmetry of equation 9.121 obvious, try substituting 
).
. Utilizing Gauss's identity (see equation 9.120 above and [3], equation 15.1.1) provides further simplification in theorem 15a for cases 15a.2 and 15a.4. These simplifications are due to simplifications appearing in 
and 
respectively. The choice of the form of the results presented was made considering the simplicity of the results and consistency between the results.