Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of such representations have been proposed. Concentrating on Riemann-Liouville integrals whose order is in (0,1), we here present a general approach that comprises most of these variants as special cases and that allows a detailed investigation of the analytic properties of each variant. The availability of this information allows to choose concrete numerical methods for handling the representations that exploit the specific properties, thus allowing to construct very efficient overall methods.

翻译：分数阶微分和积分算子的扩散表示可为构造高效的近似数值算法提供便捷途径。现有文献中提出了多种此类表示的变体。本文聚焦于阶数在(0,1)范围内的黎曼-刘维尔积分，提出了一种包含上述多数变体作为特例的通用方法，并可对各变体的解析性质进行详细研究。这些解析性质的信息使得我们能够选择利用特定属性的具体数值方法来处理相关表示，从而构建出极为高效的整体算法。