  We study three operations on Riordan arrays. First, we investigate when the
sum of Riordan arrays yields another Riordan array. We characterize the $A$-
and $Z$-sequences of these sums of Riordan arrays, and also identify an analog
for $A$-sequences when the sum of Riordan arrays does not yield a Riordan
array. In addition, we define the new operations `Der' and `Flip' on Riordan
arrays. We fully characterize the Riordan arrays resulting from these
operations applied to the Appell and Lagrange subgroups of the Riordan group.
Finally, we study the application of these operations to various known Riordan
arrays, generating many combinatorial identities in the process.
