In this paper, we introduce and study a matrix analog of the five-parameter Mittag-Leffler function. We establish the absolute convergence of the series defining this function on the unit circle $|\varpi |=1$ under certain spectral conditions. Several fundamental properties are derived, including integral representations, derivative formulas, and differential recurrence relations. Furthermore, we obtain a variety of finite summation formulas for this matrix function and its related Fox-Wright matrix analog. Finally, we investigate the composition of the function with generalized fractional calculus operators introduced by Katugampola, deriving closed-form expressions for both fractional integrals and derivatives. The results presented here extend the theory of special matrix functions and contribute to the field of fractional calculus.

The purpose of this study was to investigate the oscillation criteria for nonlinear second-order neutral differential equations with deviating arguments, with a particular emphasis on their non-canonical forms. The primary goal was to expand the current theoretical framework by introducing new relations that improved the monotonicity of positive solutions. To attain this purpose, an iterative technique was used to deduce new oscillation criteria, which helped to enhance present understanding in this field. The study process was based on a thorough review of previous literature, followed by the creation of new oscillation criteria with both theoretical and applied significance. The obtained results were validated by three illustrative instances, demonstrating the importance and influence of these criteria in the study of neutral differential equations, particularly in the study of neutral differential equations, especially in nonlinear contexts.