On the method of riordan arrays

In this work, some special number arrays along with their various properties and applications are examined. In the introduction part after some general information about the subject is given, the number arrays that will be studied in this context are presented. In the second part, formal power series and formal Laurent series are defined and their basic properties are given. After that, by means of these series, the general and basic operations on the generating functions are explained. Some various arrays possessing different recurrence relations are examined by the help of generating functions. Next, using formal power series and generating functions, Riordan arrays are defined. Basic rules and theorems of this method which make it possible to be used in different areas are analysed. Finally, by using Lagrange Inversion Formula, it has been shown that many combinatoric identities can be obtained by the method of Riordan arrays. Furthermore, equations involving the relations between the elements of some special number arrays and some other arrays are expressed.