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

Abstract

Abstract The main purpose of the present article is to introduce the classes of generalized fractional order difference sequence spaces l ∞ ( Γ , Δ α , p ) , c 0 ( Γ , Δ α , p ) and c ( Γ , Δ α , p ) by defining the fractional difference operator Δ α x k = ∑ i = 0 ∞ ( - 1 ) i Γ ( α + 1 ) i ! Γ ( α - i + 1 ) x k + i , where α is a positive proper fraction and 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 N -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 ) as the Gamma function of n , defined by an improper integral Γ ( n ) = ∫ 0 ∞ e - t t n - 1 dt , where n ∉ { 0 , - 1 , - 2 , … } and Γ ( n + 1 ) = n Γ ( n ) .