This paper intends to develop a q-difference operator \(\nabla ^{(\gamma )}_q\) of fractional order \(\gamma \) , and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence spaces \(\ell _p(\nabla ^{(\gamma )}_q)\) and \(\ell _\infty (\nabla ^{(\gamma )}_q)\) , at the same time comparing these spaces with those already exist in the literature. Apart from obtaining Schauder basis, we determine \(\alpha \) -, \(\beta \) -, and \(\gamma \) -duals of the newly defined spaces. A section is also devoted for characterizing matrix classes \((\ell _p(\nabla ^{(\gamma )}_q),\mathfrak X),\) where \(\mathfrak X\) is any of the spaces \(\ell _\infty ,\) c, \(c_0\) and \(\ell _1\) .
