This chapter deals with the concepts of generating functions, recurrence relations, related theory and applications. The concepts of this chapter are defined by using examples of computer science. Enumeration problem is a fundamental category of the problem in DM and combinatorics and is solved with the help of generating functions (GF). In other words, GF is a powerful tool to solve various enumeration problems. For a given problem size n, the main objective of the enumeration problem is to determine the number of objects which satisfies certain conditions or definitions of size n. Enumeration problems are used in various applications of DM like algorithm analysis for analyzing the configuration or counting number of inputs in algorithms, predicting or deciding the number of outputs in computation using probabilistic models, possible number of in the encryption process while using encryption and exploring spanning trees using graphs for the purpose of finding paths, cycles, subtrees, etc. The chapter concludes with case studies and projects aimed at connecting theoretical relational structures to real-world applications across various disciplines.