A note on paranormed difference sequence spaces of fractional order and their matrix transformations

In the present paper, new classes of difference sequence spaces X(Γ, Δα, u, p) for X ∈ {ℓ∞, c, c0} are introduced by using the fractional difference operator Δα, defined by Δα(xk)=∑i=0∞(-1)iΓ(α+1)i!Γ(α-i+1)xk+i, where (un) is a sequence satisfying certain conditions and α is a proper fraction. In fact, the operator Δα generalizes the difference operators used in several difference sequence spaces such as X(Δ), X(Δ2), Δm(X), ΔX(p), ΔmX(p), X(u; Δ2), X(u, Δ, p), X(u, Δ2, p) (see [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [20], [21] and [22]). Also, we investigate the topological structures and establish α−, β− and γ− duals of the spaces X(Γ, Δα, u, p). Furthermore, the matrix transformations between these spaces and the basic sequence spaces ℓ∞(q), c0(q) and c(q) are characterized.