Deferred weighted statistical and modular convergence generated by admissible lower triangular transformations: fractional $q$-difference applications

We studied deferred weighted statistical and modular convergence generated by admissible lower triangular transformations. Let $A=(a_{nk}{)}_{n,k\ge 0}$ be a triangle with nonzero diagonal entries, uniformly bounded absolute row sums, and null columns. Convergence is defined through the transformed sequence $Ax$ along admissible deferred weighted windows. We proved that the column-null condition is equivalent to $A(\phi )\subseteq c_0$, where $\phi$ denotes the space of finitely supported sequences, and use this characterization to obtain stability under finite modifications. We established uniqueness, linearity, strong-to-statistical implications, bounded converses, Cauchy characterizations, ideal and Musielak–Orlicz extensions, and window-comparison results. Analogous results are obtained for separable transformations of double sequences. As an application, we showed that the fractional $q$-difference triangle $Q^{(q,\xi )}$ has an ${\ell }_1$ coefficient kernel, uniformly bounded absolute row sums, and null fixed columns. Integer orders yield banded triangles, whereas noninteger orders produce nonterminating but absolutely summable kernels.