The Riordan Group and Applications

Chapter 1 provides a historical context for the development of Riordan arrays, using sequence-based examples, along with some tree-based applications. Chapter 2 gives an overview of some of the principal tools used in working with Riordan arrays. In Chap. 3, the formal definition of Riordan arrays and the Riordan group is addressed. In Chap. 4, useful characterizations of Riordan arrays are given. The application of Riordan array techniques to combinatorial sums and combinatorial inversions is explored in Chap. 5. As an active field of research, there have been many developments in the area of Riordan arrays since their original introduction. Chapter 6 begins the exposition of some of these, under the banner of generalized Riordan arrays. The next chapter, Chap. 7, presents a further generalization of the Riordan group. The constituent arrays of this generalization are now three-dimensional arrays, and this new structure provides a group extension of the Riordan group. Given the applications of Riordan arrays to combinatorial problems, it is natural to ask what a q-analog of a Riordan array would look like. This question, the q-analog of Riordan arrays, is studied in Chap. 8. The final chapter of the book, Chap. 9, is concerned with the question: what orthogonal polynomial sequences have Riordan arrays as their coefficient arrays?