从单项式上的半阶递推到一般分数阶微分：函数延拓与算子合成
From the Half-Order Recurrence to General Fractional-Order Differentiation on Monomials: Functional Continuation and Operator Composition
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中文总结 AI 辅助
本研究基于离散半阶系数结构，构造单项式上一般分数阶微分算子，经验证其与Riemann-Liouville公式一致，明确了函数延拓等的作用，未引入新的经典分数阶导数。
中文摘要 AI 辅助
本研究基于第一部分建立的离散半阶系数结构，开发了单项式上的一般阶分数阶微分算子。从整数系数族的一步传递律出发，将离散系数关系延拓至实自变量，在给定的归一化与正则性要求下选取了规范的Gamma函数形式。随后通过将一阶导数划分为相等的算子步长，将算子阶数扩展至半阶以上，从阶数1/m和k/m推广至一般有理阶与实阶。所得系数通过单项式系统上算子阶数的加法律独立验证，验证条件为合成过程中所有中间表达式均有定义。在对应参数域上，该构造得到了分数阶微分的标准Gamma比率单项式公式；对于非负整数阶，其退化为普通的重复微分；对于正非整数阶，在单项式层面与下限为0的左侧Riemann-Liouville公式一致。研究还讨论了其与Caputo算子的关系，包括在常数项与低次多项式项上的区别。本研究的目的并非引入新的经典分数阶导数，而是提供一条从离散系数定律到一般分数阶单项式公式的构造性代数-算子路径，阐明函数延拓、Gamma归一化与算子合成的作用。
英文摘要
This work develops a general-order fractional differentiation operator on monomials from the discrete half-order coefficient structure established in Part I. Starting from the one-step transfer law of the integer coefficient family, the discrete coefficient relation is continued to a real argument and a canonical Gamma-functional form is selected under the stated normalization and regularity requirements. The operator order is then extended beyond one half by dividing the first derivative into equal operator steps, leading from orders 1/m and k/m to general rational and real orders. The resulting coefficient is independently verified through the addition law for operator orders on the monomial system, subject to the requirement that all intermediate expressions in the composition are defined. On the corresponding parameter domain, the construction yields the standard Gamma-ratio monomial formula for fractional differentiation. For nonnegative integer orders it reduces to ordinary repeated differentiation, while for positive non-integer orders it agrees at the monomial level with the left-sided Riemann-Liouville formula with lower limit 0. Its relation to the Caputo operator is also discussed, including the distinction for constants and low-degree polynomial terms. The aim is not to introduce a new classical fractional derivative, but to provide a constructive algebraic-operator route from a discrete coefficient law to the general fractional-order monomial formula, clarifying the roles of functional continuation, Gamma normalization, and operator composition.
