The method of almost convergence with operator of the form fractional order and applications

The purpose of this paper is twofold. First, basic concepts such as Gamma function, almost convergence, fractional order difference operator and sequence spaces are given as a survey character. Thus, the current knowledge about those concepts are presented. Second, we construct the almost convergent spaces with fractional order difference operator and compute dual spaces which help us in the characterization of matrix mappings. After we characterize to the matrix transformations, we give some examples. In this paper, the notation Γ ( n ) will be shown the Gamma function. For n ∉ { 0 , − 1 , − 2 , . . . } , Gamma function is defined by an improper integral Γ ( n ) = ∫ ∞ 0 e ^− t t ^n − 1 dt .