  Asymptotic Pade-approximant methods are utilized to estimate the order
$\alpha_s^5$ contribution to the $H\to gg$ rate and the order $\alpha_s^4$
contribution to the $H \to b \bar{b}$ rate. The former process is of particular
interest because of the slow convergence evident from the three known terms of
its QCD series, which begins with an order $\alpha_s^2$ leading-order term. The
order $\alpha_s^5$ contribution to the $H \to gg$ rate is expressed as a
degree-3 polynomial in $L \equiv ln (\mu^2 / m_t^2 (\mu))$. We find that
asymptotic Pade-approximant predictions for the coefficients of $L$, $L^2$, and
$L^3$ are respectively within 1%, 2%, and 7% of true values extracted via
renormalization-group methods. Upon including the full set of next-order
coefficients, the $H \to gg$ rate is found to be virtually scale-independent
over the 0.3 $M_{H} < \mu < M_t$ range of the renormalization scale-parameter
$\mu$. We conclude by discussing the small order $\alpha_s^4$ contribution to
the $H \to b \bar{b}$ rate, which is obtained from a prior asymptotic
Pade-approximant estimate of the order $\alpha_s^4$ contribution to the
quark-antiquark scalar-current correlation function.
