Difference Sequence Spaces Derived by using Pascal Transform

The essential goal of this manuscript is to investigate some novel sequence spaces of $p_{\infty }\left( \Delta \right) $, $p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $ which are comprised by all sequence spaces whose differences are in Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$, respectively. Furthermore, we determine both $\gamma $-, $\beta $-, $\alpha $- duals of newly defined difference sequence spaces of $p_{\infty }\left( \Delta \right) $, $% p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $. We also obtain bases of the newly defined difference sequence spaces of $p_{c}\left( \Delta \right) $ and $p_{0}\left( \Delta \right) $. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes $(p_{c}\left( \Delta \right) :l_{\infty })$ and $(p_{c}\left( \Delta \right) :c)$ are characterized.