Abstract
In this article we introduce binomial difference sequence spaces of fractional order α,
$\begin{array}{}
b_p^{r,s}
\end{array}$ (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, Br,s and fractional difference operator Δ(α), defined by (Δ(α)x)k =
$\begin{array}{}
\displaystyle
\sum\limits_{i=0}^{\infty}(-1)^i\frac{\Gamma(\alpha+1)}{i!\Gamma(\alpha-i+1)}x_{k-i}
\end{array}$. 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 (
$\begin{array}{}
b_p^{r,s}
\end{array}$(Δ(α)), Y), where Y ∈ {ℓ∞, c, c0, ℓ1} and certain classes of compact operators on the space
$\begin{array}{}
b_p^{r,s}
\end{array}$(Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space
$\begin{array}{}
b_p^{r,s}
\end{array}$(Δ(α)) (1 < p < ∞).
Euler-Ces`aro difference spaces of bounded, convergent and null sequences