Generating functions containing words defined over lexicographical ordered finite alphabet and de bruijn type sequences and their applications

In this thesis, it is studied on the construction of the generating functions for the numbers counting the words defined over the finite lexicographical ordered alphabet such as Lyndon words, and also De Bruijn type sequences. With the help of the constructed generating functions, relations of these words and sequences with some special functions, the families of the Apostol type numbers and polynomials, the Stirling numbers and other special families of numbers and polynomials have been investigated. Moreover, new formulas, relations, identities and combinatorial sums including binomial coefficients and special numbers and polynomials have been obtained with the aid of the constructed generating functions and their differential equations. By using the obtained results, combinatorial sums, including binomial coefficients and the numbers of the Lyndon words having prime number length, have also been obtained. In addition to these, computational algorithms for numerical calculations are given for some of the obtained results, and related tables are given using these algorithms. The graphs of the generating functions are also drawn with the help of the values in the given tables. Finally, some applications have been given by examining relations between the obtained generating functions, prime numbers and the unique prime factorization method. Moreover, in this thesis, open problems including some of our results are given.