Bundles of acceleration on Banach manifolds

We consider an infinite-dimensional manifold M modelled on a Banach space E and we construct smooth fiber bundle structures on the tangent bundle of order two T2M, which consists of all smooth curves of M that agree up to their acceleration, as well as on the corresponding second-order frame bundle L2M. These bundles prove to be associated with respect to the identity representation of the general linear group GL(E) that serves as the structure group of both of them. Moreover, a bijective correspondence between linear connections on T2M and connection forms of L2M is revealed.