Extending the Legacy: A Journey into the Emerging World of Fractional Quantum Calculus
Keywords:
Fractional quantum calculus, Fractional difference calculus, Fractional 𝑞-difference calculus, Fractional Hahn difference calculus, Symmetric Hahn difference, (𝑝,𝑞)-calculus, Non-local phenomena, Memory effects, Discrete systemsAbstract
This paper provides a comprehensive and rigorous exposition of Fractional Quantum Calculus (FQC), a sophisticated mathematical framework that systematically extends classical integer-order quantum calculus to the realm of arbitrary fractional orders. We delineate the construction of five interconnected yet distinct types of fractional difference operators: the foundational Fractional Difference Calculus (FDC), Fractional -Difference Calculus (FqDC), and Fractional Hahn Difference Calculus (FHDC), complemented by the advanced generalizations Fractional Symmetric Hahn Difference Calculus (FSHC) and Fractional -Calculus (FpqC). The unifying principle undergirding all five frameworks involves the non-integer generalization of iterated summation, leading to the derivation of both Riemann-Liouville and Caputo operators. Fractional quantum calculus furnishes an indispensable analytical instrument for modeling intricate physical and biological phenomena characterized by inherent non-locality, memory-dependent behavior, and discrete dynamical evolution, capabilities that transcend the limitations of conventional integer-order mathematical models (see [1],[2]).
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