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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 [3–12,20–22]). 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

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