Row sums and alternating sums of Riordan arrays

Here we use row sum generating functions and alternating sum generating functions to characterize Riordan arrays and subgroups of the Riordan group. Numerous applications and examples are presented which include the construction of Girard–Waring type identities. We also show the extensions to weighted sum (generating) functions, called the expected value (generating) functions of Riordan arrays.