分数算子和特殊函数。I.贝塞尔函数
Fractional operators and special functions. I. Bessel functions
摘要：
数学物理中的大多数特殊函数都与李群的表示有关。相关李代数的元素$D$作为线性微分算子的作用给出了一类函数之间的关系，例如它们的微分递归关系。在本文中，我们定义分数推广$D^μ$这些运营商的背景下，李理论，确定其形式的属性，并说明其使用中获得有趣的功能之间的关系。我们在这里将注意力限制在欧几里德群E（2）和贝塞尔函数。我们表明，两个变量的分数算子的关系直接导致积分表示的贝塞尔函数，再现已知的分数积分为这些功能时，减少到一个变量，并有助于一个连贯的理解的连接的许多属性的功能的基本组结构。在下面的论文中，我们将分析扩展到相关的勒让德函数。
英文摘要：
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 define fractional generalizations $D^μ$ of these operators in the context of Lie theory, determine their formal properties, and illustrate their use in obtaining interesting relations among the functions. We restrict our attention here to the Euclidean group E(2) and the Bessel functions. We show that the two-variable fractional operator relations lead directly to integral representations for the Bessel functions, reproduce known fractional integrals for those functions when reduced to one variable, and contribute to a coherent understanding of the connection of many properties of the functions to the underlying group structure. We extend the analysis to the associated Legendre functions in a following paper.
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