After searching for a while, I wonder if a continued fraction representation exists for the multivariate Mills ratio $P(Z \geq x)/\phi_Z(x)$ where $Z \tilde\, N(\mu,\Sigma)$. The representation should, of course, be in terms of $x$, $\mu$, and $\Sigma$.
Such a representation is well-known if $Z \tilde\, N(0,1)$; see, for example,
(The same question is posted on stats stackexchange, see here)