Recurrent form of the renormalization group relations for the higher-order hadronic vacuum polarization function perturbative expansion coefficients

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.