标题： 分数算子和特殊函数。 II. 勒让德函数 显示英文标题

标题： Fractional operators and special functions. II. Legendre functions

摘要： 大多数数学物理的特殊函数都与李群的表示有关。关联李代数的元素 $D$ 作为线性微分算子的作用，在一类函数之间给出了关系，例如它们的微分递推关系。本文中，我们应用之前在李理论背景下发展起来的这些算子的分数阶推广 $D^\mu$，研究了 SO(2,1) 群及其共形扩展。分数阶关系为相关的勒让德函数提供了多种有趣的联系。我们证明了双变量分数阶算子关系直接导出了勒让德函数之间的积分关系，以及这些函数的一元和二元积分表示形式。某些关系在降为一元时简化为已知的勒让德函数的分数阶积分。这些结果基于潜在的群结构，扩大了对相关勒让德函数许多性质的理解。 显示英文摘要

摘要： Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a class, for example, their differential recurrence relations. In this paper, we apply the fractional generalizations $D^\mu$ of these operators developed in an earlier paper in the context of Lie theory to the group SO(2,1) and its conformal extension. The fractional relations give a variety of interesting relations for the associated Legendre functions. We show that the two-variable fractional operator relations lead directly to integral relations among the Legendre functions and to one- and two-variable integral representations for those functions. Some of the relations reduce to known fractional integrals for the Legendre functions when reduced to one variable. The results enlarge the understanding of many properties of the associated Legendre functions on the basis of the underlying group structure.