The previous three chapters discuss the partial Bell Polynomial and its application to integer sequences and various polynomials. A new formula using the Bell Polynomial to calculate Jacobi elliptic functions is described in this paper. The infinite polynomial expansion calculates the inverse of the elliptic integrals of the first and second kind. A single formula for the elliptic sine function, sn, is sufficient to calculate all 12 elliptic functions associated with the elliptic integral of the first kind. A corresponding formula for the elliptic integral of the second kind is also described.
These functions are used in various applications such as the geometry of ellipses and in astrophysics such as calculation of relativistic elliptical orbits of planets and comets. Examples of calculated elliptical and hyperbolic orbits such as the precession of the perihelion of the planet Mercury are shown using the Bell Polynomial (sn) function. An association of the Bell Polynomial function to the q octic and the Ramanujan ladder for Weber’s class invariants is discussed