Applications of derivative and difference operators in sequence and series analysis

In this work, derivative operator and difference operator, which are important tools for mathematical analysis and concrete mathematics, respectively; are applied to some families of polynomials and numbers, and some special functions. First, some families of polynomials and numbers, and some special functions which we use, are described together with their analytical and combinatorical meaning. After that, operators, which we use for analysing some families of polynomials and numbers, and some special functions, are explained by their definitions, examples and fundamental properties. Then, results which we see important in the literature, are explained in detail. In the fourth section, some general results are obtained via operators in question. And then by applying these results to some special numbers and functions, new properties and relations about these numbers are obtained. Also, new identities are obtained via operators being applied to Fibonacci, harmonic and hyperharmonic numbers. Different proofs of some known identities are also given. Besides, it is described that hyperharmonic numbers with negative order and Fibonacci numbers with negative index and explained that why their definitions are natural. Finally, digamma function which is important for both complex analysis and number theory, is analysied via operators which we mention above.