Mac Mahon partition analysis in positive integers

This master thesis is to contribute to the partition theory with new results by examining the intentional partition theory and the Mac Mahon partition analysis. This thesis consists of five parts. In the first chapter, information is given which includes the historical beginning of the partition theory. In the first part of the second chapter, discusses the partition theory and some basic problems are given without proofs. In addition, unrestricted partitions is mentioned, and geometric representations of partitions are given. Then, Jacobi triple product identity and proof, which is one of the important relativities for partition theory, are given. We mention the generating functions for the decomposition of integers, and after generating the generating function for p(n), the generating functions of Q(n) and Z(n) are given without proof. In the course of the studies, the definition of five-point numbers is given and the connection between the five-point numbers and the partition is mentioned. Euler's Pentogonal Number Theorem and also the combination of Euler's pentogonal number theorem are given. Euler's recursion formula for partition is also given in this section. The third chapter begins with the definition and properties of the Ω_{≥} operator, which has an important place in understanding the Mac Mahon partition analysis. Then, the basic features of the Mac Mahon partition analysis were studied and researched. In the fourth chapter, the Jacobi identity is given for polygonal numbers by defining the polygonal numbers. Then, the relations between the pentogonal and hexagonal numbers and the pentogonal and hexagonal numbers are examined. Some of the features of the Jacobi Triple Product Identification have been continued. Finally, new correlations between Q(n) and p(n), p(n) and Z(n) and Q(n) and Z(n) are obtained. In the last chapter, the new recurrence formulas are compared with Euler's and Ewell's recurrence formulas.