some new paranormed sequence spaces derived by q-second difference matrix

The goal of this research is to construct the extended versions of the original maddox’s paranormed sequence spaces, denoted by the notation ℓ(∇2 q, p) and ℓ∞(∇2 q, p). these spaces are linear isomorphic to the spaces ℓ(p) and ℓ∞(p), respectively. the next step is to build the schauder basis for the ℓ(∇2 q, p) space. after that, the topological features of the alpha, beta, and gamma duals of ℓ(∇2 q, p) and ℓ∞(∇2 q, p) are investigated. finally, some matrix classes are characterized.