On the classes of fractional order difference sequence spaces and their matrix transformations

The main purpose of the present article is to introduce the classes of generalized fractional order difference sequence spaces ℓ∞(Γ,Δα̃,p),c0(Γ,Δα̃,p) and c(Γ,Δα̃,p) by defining the fractional difference operator Δα̃xk=∑i=0∞(-1)iΓ(α̃+1)i!Γ(α̃-i+1)xk+i, where α̃ is a positive proper fraction and k∈N={1,2,3….}k∈N={1,2,3….}. Results concerning the linearity and various topological properties of these spaces are established and also the alpha-, beta-, gamma- and NN-duals of these spaces are obtained. The matrix transformations from these classes into Maddox spaces are also characterized. Throughout the article we use the notation Γ(n)Γ(n) as the Gamma function of n  , defined by an improper integral Γ(n)=∫0∞e-ttn-1dt, where n∉{0,-1,-2,…} and Γ(n+1)=nΓ(n)Γ(n+1)=nΓ(n).