Next-to-next-to-leading order vacuum polarization function of heavy quark near threshold and sum rules for $b\bar{b}$ system

A correlator of the vector current of a heavy quark is computed analytically near threshold in the next-to-next-to-leading order in perturbative and relativistic expansion that includes ${\text{al}}_s^2$, ${\text{al}}_{s}v$ and $v^2$ corrections in the coupling constant and velocity of the heavy quark to the nonrelativistic Coulomb approximation. Based on this result, the numerical values of the $b$-quark pole mass and the strong coupling constant are determined from the analysis of sum rules for the $\mathrm{{\rm Y}}$ system. The next-to-next-to-leading corrections are found to be of order of next-to-leading ones.