Recently, Devaiya and Srivastava [3] studied the \(T^r\) -strong convergence of numerical sequences and Fourier series using a lower triangular matrix \(T=(b_{m,n})\) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce \([T^r,G,u,q]\) -strongly convergent sequence spaces for \(r\in\mathbb{N}\) , and defined by a sequence of modulus functions. We also provide a relationship between \([T,G,u,q]\) and \([T^r,G,u,q]\) -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for \({T}^{r}\) -strong convergent sequences, statistical convergence and Fourier series using the idea of \([T^r,G,u,q]\) -strongly convergent sequence spaces.
