Multi-Level General Fractional Derivatives: Fundamental Theorems and Applications

General fractional derivatives (GFDs) generalize classical fractional operators by considering the specific kernel functions; however, extending this generalization to a multi-layered structure remains unexplored. Here, we propose a systematic analysis of the nth-level GFDs, which extend the framework developed in earlier studies. We begin by revisiting the first- and second-level GFDs, proceeding to explore the properties of general fractional integrals (GFIs) to provide a higher-level mathematical foundation. Subsequently, we examine illustrative examples and establish the first and second fundamental theorems of fractional calculus (FC) within the nth-level setting. The proof of the first fundamental theorem follows the pattern of the lower-level cases, while the second fundamental theorem is verified by deriving an inductive representation for the null-space of associated kernels. Finally, the paper concludes with an analysis of fractional differential equations (FDEs) employing GFDs.