Extension of Delaunay normalisation for arbitrary powers of the radial distance

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 ϒ. 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 Є [0,1) and E Є [-ꙥ ꙥ]. 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.