On sequence spaces generated by binomial difference operator of fractional order

Abstract

Abstract In this article we introduce binomial difference sequence spaces of fractional order α, b p r , s $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, B r,s and fractional difference operator Δ(α), defined by (Δ(α) x) k = ∑ i = 0 ∞ ( − 1 ) i Γ ( α + 1 ) i ! Γ ( α − i + 1 ) x k − i $\begin{array}{} \displaystyle \sum\limits_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i} \end{array}$ . We give some topological properties, obtain the Schauder basis and determine the α, β and γ-duals of the spaces. We characterize the matrix classes ( b p r , s $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)), Y), where Y ∈ {ℓ ∞, c, c 0, ℓ 1} and certain classes of compact operators on the space b p r , s $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space b p r , s $\begin{array}{} b_p^{r,s} \end{array}$ (Δ(α)) (1 < p < ∞).