By the help of linear recurrence relations defining generating functions of some special numbers and polynomials and their applications

In this thesis, generating functions of some special numbers and polynomials are defined with the help of linear recurrence relations and some applications are made by making use of this. With the help of generating functions, the properties of some special number and polynomial families were examined and the relations and formulas between them were studied. In this thesis, a sequence of numbers corresponding to the number of all repetitive s-permutations that can be written on the set {0, 1, 2}, provided that two zeros and two ones do not come together are discussed.With the help of the recurrence relation given by Charalambides (2002), the generating function of this number sequence has been studied and the relations of this generating function with some special functions are given.With the help of the related generating function, the solution of the homogeneous linear recurrence relation given by Problem 7.7.11 in the book of Charalambides (2002) is also given. In addition, in this thesis some obtained results have been related to a polynomial family given by Simsek (2023), especially including Fibonacci type and Lucas type number and polynomial families. At the same time, a new special polynomial family was defined with the related generating function and some basic properties of these polynomials were examined. Using some of these properties, many new formulas have been derived that can be used in mathematics and other applied sciences. By applying the derivative operator to these polynomials, new formulas are also obtained. Also, Riemann integral representations for these polynomials are given. The new results which are obtained, include some special numbers and polynomials like Stirling numbers of the first kind, Bernoulli polynomials of the second kind (Cauchy numbers), Fibonacci numbers and Pell numbers. In this thesis, a large number of comparative notes have been given about the results obtained within the scope of the thesis and interpretations have been made depending on these.