In this article we introduce binomial difference sequence spaces of fractional order α,\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,$$\end{document}b0r,sΔ(α),\documentclass[12pt]{minimal}
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\begin{document}$$b_0^{r,s}\left( \Delta ^{(\alpha )}\right) ,$$\end{document}bcr,sΔ(α)\documentclass[12pt]{minimal}
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\begin{document}$$b_c^{r,s}\left( \Delta ^{(\alpha )}\right) $$\end{document} and b∞r,sΔ(α)\documentclass[12pt]{minimal}
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\begin{document}$$b_{\infty }^{r,s}\left( \Delta ^{(\alpha )}\right) $$\end{document} by employing fractional difference operator Δ(α),\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^{(\alpha )},$$\end{document} defined by Δ(α)xk=∑i=0∞(-1)iΓ(α+1)i!Γ(α-i+1)xk-i.\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^{(\alpha )}x_k=\sum \limits _{i=0}^{\infty }(-1)^i\frac{\Gamma (\alpha +1)}{i!\Gamma (\alpha -i+1)}x_{k-i}.$$\end{document} We give some topological properties, obtain the Schauder basis and determine the α-,\documentclass[12pt]{minimal}
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\begin{document}$$\alpha -,$$\end{document}β-\documentclass[12pt]{minimal}
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\begin{document}$$\beta -$$\end{document} and γ-\documentclass[12pt]{minimal}
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\begin{document}$$\gamma -$$\end{document} duals of the spaces. We characterize the matrix classes (bcr,s(Δ(α)),ℓp),\documentclass[12pt]{minimal}
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\begin{document}$$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _p),$$\end{document}(bcr,s(Δ(α)),ℓ∞)\documentclass[12pt]{minimal}
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\begin{document}$$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _{\infty })$$\end{document} and (bcr,s(Δ(α)),c).\documentclass[12pt]{minimal}
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\begin{document}$$(b_c^{r,s}(\Delta ^{(\alpha )}),c).$$\end{document} We characterize certain classes of compact operators on the space bcr,s(Δ(α))\documentclass[12pt]{minimal}
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\begin{document}$$b_c^{r,s}(\Delta ^{(\alpha )})$$\end{document} using Hausdorff measure of non-compactness. Finally, we present the graphical interpretation of the operator Br,sΔ(α)\documentclass[12pt]{minimal}
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\begin{document}$$B^{r,s}\left( \Delta ^{(\alpha )}\right) $$\end{document}.
