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

The $\{\beta\}$-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order $O(\alpha_s^4)$