The authors define the \(\Delta^m(c_I)\) spaces by using generalized sequence spaces and ideal convergence. Furthermore they establish some topological results and give inclusion relation between \( \Delta ^{m}(w_{p}^{T})\)-convergence and \(\Delta ^{m}\)-ideal convergence. They also prove the following theorems:
Let \(I\) be an admissible ideal, and let \(x=(x_{k})\) and \( y=(y_{k})\) be real valued sequences.
(a) If \(x_{k}\rightarrow L_{1}(\Delta ^{m}(c_{I}))\) and \(y_{k}\rightarrow L_{2}(\Delta ^{m}(c_{I}))\) then \(x_{k}+y_{k}\rightarrow (L_{1}+L_{2})(\Delta ^{m}(c_{I})).\)
(b) If \(x_{k}\rightarrow L_{1}(\Delta ^{m}(c_{I}))\) and \(\lambda \in\mathbb{R}\) then \(\lambda x_{k}\rightarrow \lambda L_{1}(\Delta ^{m}(c_{I})).\)
If \(x=(x_{k})\) is a \(\Delta ^{m}I\)-convergent sequence then \(x\) is a \(\Delta ^{m}I\)-Cauchy sequence.
Let \(I\) be a non-trivial ideal in \(\mathbb{N}\) and \(x\) be a sequence. If there is a \(\Delta ^{m}I\)-convergent \(y\) such that \[ \left\{ k\in\mathbb{N}:\Delta ^{m}x_{k}\neq \Delta ^{m}y_{k}\right\} \in I \] then \(x\) is also \(\Delta ^{m}I\)-convergent.
Let \(p\in\mathbb{R}\), \(0<p<\infty\), \(T=(t_{nk})\) be a non-negative regular matrix and \(A\subseteq\mathbb{N}.\) If \(x_{k}\rightarrow L_{1}(\Delta ^{m}(w_{p}^{T}))\) then \(x_{k}\rightarrow L(\Delta ^{m}(c_{I_{d_{T}}})).\) If \(x\in \Delta ^{m}(l_{\infty })\) and \(x_{k}\rightarrow L(\Delta ^{m}(c_{I_{d_{T}}}))\) then \(x_{k}\rightarrow L(\Delta ^{m}(w_{p}^{T})).\)