We introduce a generalized quantum (q, ) -fractional complex step framework for the numerical evaluation of fractional derivatives based on quantum-deformed special functions. We develop rigorous definitions of the (q, ) -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, ) -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.
Ibrahim et al. (Thu,) studied this question.
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