The Multiple Riordan Group and the Multiple Riordan Semigroup

Abstract: In this paper, we define Riordan type arrays and the Riordan semigroup and extend them to multiple settings. Multiple Riordan type arrays and the multiple Riordan semigroup are related to multiple Riordan arrays and the multiple Riordan group. We give a comprehensive discussion of the latter and characterize them by an $A$-sequence and multiple $Z$-sequences. Applications of Riordan type arrays in the construction of identities are given. In addition, we give compressions of multiple Riordan arrays and multiple Riordan type arrays and their sequence characterizations.