  We calculate the local Green function for a quantum-mechanical particle with
hopping between nearest and next-nearest neighbors on the Bethe lattice, where
the on-site energies may alternate on sublattices. For infinite connectivity
the renormalized perturbation expansion is carried out by counting all
non-self-intersecting paths, leading to an implicit equation for the local
Green function. By integrating out branches of the Bethe lattice the same
equation is obtained from a path integral approach for the partition function.
This also provides the local Green function for finite connectivity. Finally, a
recently developed topological approach is extended to derive an operator
identity which maps the problem onto the case of only nearest-neighbor hopping.
We find in particular that hopping between next-nearest neighbors leads to an
asymmetric spectrum with additional van-Hove singularities.
