  In a previous paper we performed an analysis of asymptotic structure of
perturbation theory series for semileptonic $\tau$-lepton decays in massless
limit. We extend our analysis to the Cabibbo suppressed $\Delta S=1$ decay
modes of the $\tau$ lepton. In particular we address the problem of $m_s^2$
corrections to theoretical formulas. The properties of the asymptotic behavior
of the finite order perturbation theory series for the coefficient functions of
the $m_s^2$ corrections are studied.
