Adler function and Bjorken polarized sum rule: Perturbation expansions in powers of $SU(N_c)$ conformal anomaly and studies of the conformal symmetry limit

We consider a new form of analytical perturbation theory expansion in the massless $SU(N_c)$ theory, for the non-singlet part of the $e^+e^-$-annihilation to hadrons Adler function $D^{ns}$ and of the Bjorken sum rule of the polarized lepton-hadron deep-inelastic scattering $C_{ns}^{Bjp}$, and demonstrate its validity at the $O(\alpha_s^4)$-level at least. It is a two-fold series in terms of powers of the conformal anomaly and of $SU(N_c)$ coupling $\alpha_s$. Explicit expressions are obtained for the $\{\beta\}$-expanded perturbation coefficients at $O(\alpha_s^4)$ level in $\bar{\rm MS}$ scheme, for both considered physical quqantities. Comparisons of the terms in the $\{\beta\}$-expanded coefficients are made with the corresponding terms obtained by using extra gluino degrees of freedom, or skeleton-motivated expansion, or $R_{\delta}$-scheme motivated expansion in the Principle of Maximal Conformality. Relations between terms of the $\{\beta\}$-expansion for the $D^{ns}$- and $C_{ns}^{Bjp}$-functions, which follow from the conformal symmetry limit and its violation, are presnted. The relevance to the possible new analysis of the experimental data for the Adler function and Bjorken sum rule is discussed.