Title: Fractional operators and multi-integral representations for associated Legendre functions Show Chinese title

Title: 分数阶算子和关联勒让德函数的多重积分表示

Abstract: In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps. These are of potential use in a number of physical problems. We show here how their results can be derived simply from more general relations involving non-integer changes in the order obtained using the fractional group operator methods developed earlier for SO(2,1), E(2,1) and its conformal extension, and SO(3). We also present general integral relations for fractional changes of the degrees of the functions, and related multi-derivative and multi-integral representations. Show Chinese abstract

Abstract: 在最近的一篇论文中，Cohl 和 Costas-Santos 推导出了一些关于关联勒让德函数和费雷尔函数的有趣多导数和多积分关系，其中这些函数的阶数以整数步长发生变化。这些关系在一些物理问题中可能具有潜在的应用价值。我们在这里展示了如何从涉及非整数阶数变化的更一般关系中简单地推导出他们的结果，这些关系是使用之前为 SO(2,1)、E(2,1) 及其共形扩展和 SO(3) 开发的分数群算子方法得到的。我们还提出了函数度数的分数变化的一般积分关系，以及相关的多导数和多积分表示。