Kerr spacetime and scalar wave equation: Exact resummation of the renormalized angular momentum in the eikonal limit

We show that the null geodesic radial action for unbound orbits in the Kerr spacetime, and consequently the scattering angle, can be resummed in terms of hypergeometric functions, extending previous results [Ivanov et al., Resummation of universal tails in gravitational waveforms, arXiv:2504.07862.]. We provide explicit expressions as series expansions in powers of the Kerr rotational parameter up to the fourth order included.We finally use the Mano-Suzuki-Takasugi formalism to prove the relation between the renormalized angular momentum and the radial action highlighted in previous works.