Some new difference sequence spaces of fractional order and their dual spaces

The main objective of the present paper is to introduce certain difference sequence spaces of fractional order and investigate their topological structures as well as some interesting results concerning the fractional difference operator Δα. In this paper, we define some new difference sequence spaces such as ℓ∞(Γ,Δα,u),c0(Γ,Δα,u) and c(Γ,Δα,u) by introducing a fractional difference operator Δα and for a positive fraction α, the difference operator Δα is defined by Δα(xk)=∑i=0∞(-1)iΓ(α+1)i!Γ(α-i+1)xk+i. Also we establish their α-,β- and γ-duals.