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

Abstract

We studied deferred weighted statistical and modular convergence generated by admissible lower triangular transformations. Let A = (ank)n,k >= 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) subset of c0, 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.