Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables θ _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularise highly oscillating perturbation force of some orbital systems will be explored for the first four categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, numerical results are included to provide test examples for constructing computational algorithms.