  We study the time-momentum representation of the kernel needed to compute
hadronic vacuum polarization contributions to the muon $g$-$2$ in the
space-like region at next-to-leading order. For small values of the time, we
present analytical series expansions; for large values of the time, we present
numerical series expansions which overcome the problems showed by na\"ive
asymptotic expansions. These results are to be employed in lattice QCD
determination of hadronic vacuum polarization contributions to the muon $g$-$2$
at next-to-leading order.
