In the framework of perturbed Keplerian systems we deal with the Delaunay normalisation of a wide class of perturbations such that the radial distance is raised to an arbitrary real number γ. The averaged function is expressed in terms of the Gauss hypergeometric function 2F1 whereas the associated generating function is the so called Appell hypergeometric function F1. The Gauss hypergeometric function related to the average depends on the eccentricity, e, whereas the Appell function depends additionally on the eccentric anomaly, E, and both special functions are properly defined and evaluated for all e∈[0,1) and E∈[−π,π]. We analyse when the functions we determine can be extended to e=1. When the exponent of the radial distance is an integer, the usual values of the averaged and generating functions are recovered.

中文翻译：

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径向距离任意幂的 Delaunay 归一化扩展


在扰动开普勒系统的框架中，我们处理了一大类扰动的 Delaunay 归一化，使得径向距离γ提高到任意实数。平均函数用高斯超几何函数 2F1 表示，而相关的生成函数是所谓的阿佩尔超几何函数 F1。与平均值相关的高斯超几何函数取决于偏心率 e，而阿佩尔函数还取决于偏心率异常 E，并且这两个特殊函数都针对所有 e∈[0,1） 和 E∈[−π，π] 进行了正确定义和计算。我们分析了我们确定的函数何时可以扩展到 e=1。当径向距离的指数为整数时，将恢复 averaged 和 generating 函数的通常值。