Unified framework of four Caputo fractional differences for initial and final value problems in discrete fractional calculus with variable bounds

Fractional calculus has emerged as a powerful tool for characterizing non-classical dynamic phenomena, yet its discretization remains fragmented, with existing studies primarily focusing on single combinations of variable bounds and difference directions. To address this gap, this paper proposes a unified theoretical framework for discrete fractional calculus by systematically introducing four novel Caputo fractional difference definitions, which integrate variable upper/lower-limit sums with forward/backward difference operations. First, we rigorously derive the fundamental properties of these four definitions, including the commutativity of fractional sums and differences, and their consistency with integer-order difference operations. Second, we construct fractional difference equations for each definition, establish their equivalence to Volterra sum equations, and provide explicit solutions and strict proofs for their corresponding initial and final value problems. To validate the theoretical results, we design four targeted computational cases and numerical simulations, confirm the consistency between theoretical solutions and numerical results, and intuitively demonstrate the long-memory effect of fractional-order discrete systems. Furthermore, we present a concise comparison of the four definitions, clarifying their suitability for discrete systems with distinct boundary conditions and dynamic characteristics. This work not only completes the theoretical system of discrete fractional calculus with variable bounds but also provides standardized and targeted mathematical tools for modeling complex discrete dynamic processes, laying a solid foundation for the practical application of discrete fractional calculus in fields such as engineering control, infectious disease modeling, and economic dynamics.