We studied deferred weighted statistical and modular convergence generated by admissible lower triangular transformations. Let $ A = (a_{nk})_{n, k\geq0} $ 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(\varphi)\subseteq c_0 $, where $ \varphi $ 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.
Citation: Gui-Ying Weng, Ömer Kişi, Mehmet Gürdal, Qing-Bo Cai. Deferred weighted statistical and modular convergence generated by admissible lower triangular transformations: fractional $ q $-difference applications[J]. AIMS Mathematics, 2026, 11(7): 23404-23436. doi: 10.3934/math.2026944
We studied deferred weighted statistical and modular convergence generated by admissible lower triangular transformations. Let $ A = (a_{nk})_{n, k\geq0} $ 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(\varphi)\subseteq c_0 $, where $ \varphi $ 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.
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Gui-Ying Weng, Ömer Kişi, Mehmet Gürdal, Qing-Bo Cai. Deferred weighted statistical and modular convergence generated by admissible lower triangular transformations: fractional $ q $-difference applications[J]. AIMS Mathematics, 2026, 11(7): 23404-23436. doi: 10.3934/math.2026944
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