Time-kernel for lattice determinations of NLO hadronic vacuum polarization contributions to the muon $g$-$2$

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 naïve asymptotic expansions. These results are to be employed in lattice QCD determinations of hadronic vacuum polarization contributions to the muon $g$-$2$ at next-to-leading order.