  The renormalization group relations for the higher-order hadronic vacuum
polarization function perturbative expansion coefficients are studied. The
folded recurrent and unfolded explicit forms of such relations are obtained.
The explicit expression for the coefficients, which incorporate the
contributions of the $\pi^{2}$-terms in the perturbative expansion of the
R-ratio of electron-positron annihilation into hadrons, is derived at an
arbitrary loop level. The obtained results can be employed as an independent
crosscheck of the higher-order perturbative calculations of the hadronic vacuum
polarization function and in the studies of the renormalization scale setting
in the relevant physical observables.
