Some New Paranormed Sequence Spaces and $\alpha -$ Core

In this study, we define new paranormed sequence spaces by combining a double sequential band matrix and a diagonal matrix. Furthermore, we compute the $\alpha -,\beta -$ and $\gamma -$ duals and obtain bases for these sequence spaces. Besides this, we characterize the matrix transformations from the new paranormed sequence spaces to the spaces $c_0(q),c(q),\mathrm{\ell }(q)$ and ${\mathrm{\ell }}_{\mathrm{\infty }}(q)$. Finally, $\alpha -core$ of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.