  A newly developed hypergeometric resummation technique [H. Mera et al., Phys.
Rev. Lett. 115, 143001 (2015)] provides an easy-to-use recipe to obtain
conserving approximations within the self-consistent nonequilibrium many-body
perturbation theory. We demonstrate the usefulness of this technique by
calculating the phonon-limited electronic current in a model of a
single-molecule junction within the self-consistent Born approximation for the
electron-phonon interacting system, where the perturbation expansion for the
nonequilibrium Green function in powers of the free bosonic propagator
typically consists of a series of non-crossing \sunset" diagrams.
Hypergeometric resummation preserves conservation laws and it is shown to
provide substantial convergence acceleration relative to more standard
approaches to self-consistency. This result strongly suggests that the
convergence of the self-consistent \sunset" series is limited by a branch-cut
singularity, which is accurately described by Gauss hypergeometric functions.
Our results showcase an alternative approach to conservation laws and
self-consistency where expectation values obtained from conserving perturbation
expansions are \summed" to their self-consistent value by analytic continuation
functions able to mimic the convergence-limiting singularity structure.
