Statistical convergence of sequences defined by the fractional difference operator

This thesis consists of six chapters. In the first chapter, the history of the emergence of the concept of statistical convergence is mentioned, and brief information is given about the authors' determinations on this subject, its application to some spaces and its generalization. Then, the difference sequence space and its generalization, lacunary convergence, Cesáro summability, Orlicz function and generalized fractional difference operator are given. In the second chapter, some definitions, theorems and relations between them are given to be used in the thesis. In the third chapter, definition of statistical convergence, natural density and properties, Cesàro summability of a sequence and some basic properties and relations between them are mentioned. Then, the lacunary sequence, the relations between lacunary convergence and Cesàro summability, ∆^m-statistical convergence and Cesáro summability and some coverage relations are examined. In the fourth chapter, by giving the definition of Orlicz function,, c(∆^m,M,p),c_0 (∆^m,M,p),〖 l〗_∞ (∆^m,M,p), S(∆^m,M,p) and S_0 (∆^m,M,p) difference sequence spaces are given and some topological properties between these spaces are given. In the fifth chapter, given the fractional order ∆^α difference sequence space, we examined some applications, statistical convergence according to the 𝑓 modulus function and Cesàro summability, lacunary statistical convergence of fractional difference sequences, generalized fractional order ∆^α difference sequence spaces defined by Orlicz functions. In the sixth and last chapter, ∆_v^α -lacunary statistical convergence of fractional difference sequences, ∆_v^α -lacunary statistical convergence of fractional difference sequences according to 𝑓 modulus function, and ∆_v^α-Cesàro summable difference sequence spaces and fractional difference sequence spaces defined by Orlicz functions. We examined the generalized lacunary difference sequence spaces and statistical convergence of difference sequences.