  We consider an asymptotically free vectorial SU($N_c$) gauge theory with
$N_f$ massless fermions in a representation $R$, having an infrared fixed point
(IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal
non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental,
adjoint, and symmetric rank-2 tensor representation are considered. We present
scheme-independent calculations of the anomalous dimension
$\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to
$O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion
parameter. Comparisons are made with conventional $n$-loop calculations and
lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we
calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$
SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to
this order, with the exactly known expression. The $\Delta_f$ expansion also
avoids a problem in which an IRFP may not be manifest as an IR zero of a higher
$n$-loop beta function.
