  We reexamine genus one super-Green functions with general boundary conditions
twisted by $(\alpha, \beta)$ for $(\sigma, \tau)$ directions in the eigenmode
expansion and derive expressions as infinite series of hypergeometric
functions. Using these, we compute one-loop superstring amplitudes with
non-maximal supersymmetry, taking an example of massless vector emissions of
open string type I ${\cal Z}_2$ orbifold.
