  We study a vectorial asymptotically free gauge theory, with gauge group $G$
and $N_f$ massless fermions in a representation $R$ of this group, that
exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling
$\alpha=\alpha_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$,
we calculate the scheme-independent series expansions of (i) the anomalous
dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$, to
$O(\Delta_f^4)$ and (ii) the derivative $\beta' = d\beta/d\alpha$, to
$O(\Delta_f^5)$, both evaluated at $\alpha_{IR}$, where $\Delta_f$ is an
$N_f$-dependent expansion variable. These are the highest orders to which these
expansions have been calculated. We apply these general results to theories
with $G={\rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric
and antisymmetric rank-2 tensor representations. It is shown that for all of
these representations, $\gamma_{\bar\psi\psi,IR}$, calculated to the order
$\Delta_f^p$, with $1 \le p \le 4$, increases monotonically with decreasing
$N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$.
Comparisons of our scheme-independent calculations of
$\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ are made with our earlier higher
$n$-loop values of these quantities, and with lattice measurements. For $R=F$,
we present results for the limit $N_c \to \infty$ and $N_f \to \infty$ with
$N_f/N_c$ fixed. We also present expansions for $\alpha_{IR}$ calculated to
$O(\Delta_f^4)$.
