On the gauss hyper-fibonacci and gauss hyper-lucas numbers

This thesis is structured in four main chapters. In the first chapter, Fibonacci and Lucas number sequences are discussed and Binet formulas, generator functions and basic properties of these sequences are analyzed comprehensively. In addition, polynomial expressions of these numbers and certain properties of these expressions are evaluated in this section. In the second part, the definitions of Gauss Fibonacci and Gauss Lucas sequences are presented, focusing on structural properties of these numbers such as generating functions, sums of their first terms and Cassini identities. In addition, the polynomials of these sequences of numbers and their prominent properties are discussed in detail. In the third chapter, we focus on the Hyper-Fibonacci and Hyper-Lucas sequences. The recurrence relations of these numbers are explained and their relations with the classical Fibonacci and Lucas sequences are analyzed. Moreover, the generating functions of these numbers are analyzed and the basic characteristic structures of the related polynomials are presented. In the last section, Gauss Hyper-Fibonacci and Gauss Hyper-Lucas numbers are defined. The generating functions of these numbers, the properties of the sums and differences of these numbers with each other and the recurrence relations are defined.