The Riordan Group and Applications

Foreword
Preface
Part I: Lou Shapiro’s
Part II: Renzo Sprugnoli’s
Part III: Overview of the book
Acknowledgements
Contents
Notation
1 Introduction
1.1 What are Riordan Arrays?
1.2 Origins and Motivation
1.3 Elementary Applications
Exercises
References
2 Extraction of Coefficients and Generating Functions
2.1 Formal Power Series
2.2 Coefficient Extraction
2.3 Lagrange Inversion Theorem
2.4 Generating Functions
Exercises
References
3 The Riordan Group
3.1 Riordan Arrays and the Riordan Group
3.2 Some Special Subgroups
3.3 Several Aspects of the Riordan Group
Exercises
References
4 Characterization of Riordan Arrays by Special Sequences
4.1 The A- and Z- Sequences
4.2 The A-matrix
4.3 Is It a Riordan Array?
Exercises
References
5 Combinatorial Sums and Inversions
5.1 Combinatorial Sums
5.2 Combinatorial Inversions
Exercises
References
6 Generalized Riordan Arrays
6.1 Exponential Riordan Arrays
6.2 Generalized Riordan Arrays and the Riordan Group
6.3 Relations Between Riordan Arrays and Sheffer Sequences
6.4 Special Riordan Arrays and Sheffer Sequences
6.5 Double Riordan Arrays and Sheffer Polynomial Pairs
References
7 Extensions of the Riordan Group
7.1 Three-Dimensional Riordan Group
7.2 Three-Dimensional Riordan Arrays
7.3 The Riordan Group in Several Variables
Exercises
References
8 q-Analogs of Riordan Arrays
8.1 Combinatorial q-Analogs
8.2 Eulerian Generating Functions
8.3 q-Riordan Arrays
8.4 Combinatorial Applications of the q-Riordan Arrays
References
9 Orthogonal Polynomials
9.1 Orthogonal Polynomials and Riordan Arrays
9.2 Exponential Riordan Arrays and Classical Orthogonal Polynomials
9.3 Orthogonal Polynomials as Moments
9.4 Combinatorial Polynomials as Moments of Riordan Arrays
9.5 Continued Fractions and Riordan Arrays
Exercises
References
Appendix Solutions
References
Index