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

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.