**Thesis Type:** Doctorate

**Institution Of The Thesis:** Uludağ Üniversitesi, Turkey

**Approval Date:** 2017

**Thesis Language:** Turkish

**Student:** ELİF ÇETİN

**Supervisor: **İSMAİL NACİ CANGÜL

In this thesis, some new finite sums are defined and the relations between these new sums and other known finite sums are given. Besides, with the help of the PDE of the three-term polynomial relation, some finite sums are obtained. This thesis consists of five chapters. First chapter is introduction. In the second chapter some basic definitions and theorems which will be used in the other chapters are given. The third chapter contains the original results. In this chapter, some finite special sums are given and the relations between these sums and the PDE of the three-term polynomial relations are investigated. The fourth section contains the original results too. In this section, two new finite sums are defined and the relations between these new sums and the other known finite sums are investigated. In the last chapter, relations between Fibonacci numbers and the new finite sums which are defined in chapter four, are investigated. And for some specific values of Fibonacci numbers, some relations of these sums are given. Some original results are obtained.