The Generalized quantum difference operator in $c$ and $c_0$ and their duals

In this article, we present a generalization of the $k^{th}$ order of forward difference operator $\Delta^k$ using its quantum analog $\Delta^k_q$. We study its domains $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ in the spaces $c$ and $c_0$ of convergent and null sequences, respectively. Additionally, we show that the domains $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ are $BK$-spaces and linearly isomorphic to $c$ and $c_0$, respectively. Furthermore, we construct Schauder bases and examine the K"{o}the duals of the spaces $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$. The final segment deals with the characterization of certain class of matrix mappings from the spaces $c(\Delta^k_q)$ and $c_0(\Delta^k_q)$ to the space $\nu\in\{c, c_0, \ell_\infty,~\ell_1\}$.