Gaussian Chebyshev polynomials and their properties

In this thesis, by extending the well-known first, second, third and fourth kind Chebyshev polynomials to the complex plane, new families of Gaussian polynomials called Gaussian Chebyshev polynomials are defined. The basic properties of these newly defined concepts (Binet formulas, generating functions, combinatorial properties, etc.), their relations with known Chebyshev polynomials, and the relations with special number sequences and polynomials of special number sequences are included. This thesis consists of six chapters. The first chapter is the introduction and provides general information about the subject and brief information about the studies in the literature. In the second chapter, definitions and theorems related to special number sequences, polynomials of special number sequences and Chebyshev polynomials used throughout the thesis are given. The third, fourth and fifth chapters of the thesis consist of original studies. In the third chapter, the first and second kinds of Gaussian Chebyshev polynomials are defined, Binet formulas, generating formulas, their basic identities such as Cassini, Catalan, d'Ocagne, their relations with known Chebyshev polynomials, and relations with each other are examined in detail. In the fourth section, the third and fourth kinds of Gaussian Chebyshev polynomials are defined, their basic properties, their relations with known Chebyshev polynomials and first and second kind Gaussian Chebyshev polynomials are given. In the fifth section, the relationships between the newly defined four kinds of Gaussian Chebyshev polynomials and Fibonacci, Lucas, Pell and Pell-Lucas number sequences/polynomials and relationships with some special values are given. In the sixth chapter, the thesis is completed by with the conclusion and recommendation chapter.