Recurrence formulae for the Hansen's developments

Abstract

In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions ( r/a ) n cos mv and ( r/a ) n sin mv in terms of the eccentric anomaly E or the mean anomaly M . We also establish a recurrence process for computing the series expansions for all n and m when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.