Analysis of some special sequences with combinatorial inverse transforms

Finite sums, which include important number types such as harmonic numbers and their generalizations, Stirling numbers of the first and second kinds and binomial coefficients, appear in many fields such as analysis, number theory and combinatorics. Evaluating these sums or obtaining other useful forms is therefore important. There are various techniques for research in this direction, we will mainly use three of these techniques together in this thesis. First one is the inverse transformations, the most famous of which are the binomial and Stirling transforms, second one is the series transformation formulas for establishing useful relations between generating functions and the sums in question and the third one is the difference operators. Classic and current results involving these techniques were brought together and a compact reference was created. In addition to the important results in the literature, many new formulas and equations were obtained.