Theory of generating functions and their applications

Abstract

The generating functions are important instruments for solving the enumerative problems in combinatorial analysis and in number theory. Enumerative problems arise when we need to be explicit about the number of ways of choosing particular elements from a finite set. The application of generating functions in this situation consists of establishing a correspondence between the elements of the set and the terms of the products of some series; the solution of enumerative problem is reduced, in fact, to finding a suitable method for the multiplication of these series.The method of generating functions can be effectively applied to enumerative problems of graph theory, that is, problems arising when counting graphs with specific properties. In number theory, the generating functions can be used to prove some identities. In this thesis, we understand the benefits of the generating functions and discuss many identities that come from 'Partitions of Integers', and 'Stirling Numbers'. We see how we can easily prove these identities by using generating functions.