(Quantum, Deformed)-Fractional Complex Steps: Theory, Entropy Analysis, and Imaging Applications

We introduce a generalized quantum (q, τ)-fractional complex step framework for the numerical evaluation of fractional derivatives based on quantum-deformed special functions. We develop rigorous definitions of the (q, τ)-Fractional Complex Step Method (FCSM), establish convergence and error estimates under analytic regularity, and derive fractional entropy identities, including Tsallis and Shannon H-theorems, for generalized diffusion systems. Analytical properties of the (q, τ)-Gamma function and related operators are presented, together with illustrative numerical examples. The proposed framework extends classical complex step methods and provides a mathematically rigorous toolset for analyzing fractional operators in both pure and applied contexts.