On The Difference Sequence Space $l_p(\hat{T}^q)$

In this study, we introduce a new matrix $\hat{T}^q=(\hat{t}^q_{nk})$ by \[ \hat{t}^q_{nk}=\left \{ \begin{array} [c]{ccl}% \frac{q_n}{Q_n} t_n & , & k=n\\ \frac{q_k}{Q_n}t_k-\frac{q_{k+1}}{Q_n} \frac{1}{t_{k+1}} & , & k n . \end{array} \right. \] where $t_k>0$ for all $n\in\mathbb{N}$ and $(t_n)\in c\backslash c_0$ . By using the matrix $\hat{T}^q$ , we introduce the sequence space $\ell_p(\hat{T}^q)$ for $1\leq p\leq\infty$ . In addition, we give some theorems on inclusion relations associated with $\ell_p(\hat{T}^q)$ and find the $\alpha$ -, $\beta$ -, $\gamma$ - duals of this space. Lastly, we analyze the necessary and sufficient conditions for an infinite matrix to be in the classes $(\ell_p(\hat{T}^q),\lambda)$ or $(\lambda,\ell_p(\hat{T}^q))$ , where $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$ .