  We derive explicit expressions for the elements of the $\{ \beta
\}$-expansion for the nonsinglet Adler $D_A$-function and Bjorken polarized sum
rules $S^{Bjp}$ in the N$^4$LO using recent results by Chetyrkin for these
quantities computed within extended QCD including any number of fermion
representations. We discuss the properties of the $\{ \beta \}$-expansion for
$D_A$ and $S^{Bjp}$ at higher orders which follow from the Crewther [1] and the
Broadhurst-Kataev [2] relation.
