Computational investigation of static multipole polarizabilities and sum rules for ground-state hydrogen-like ions

High precision multipole polarizabilities, ${\alpha }_{\mathrm{\ell }}$ for $\mathrm{\ell }\le 4$ of the $1s$ ground state of the hydrogen isoelectronic series are obtained from the Dirac equation using the B-spline method with Notre Dame boundary conditions. Compact analytic expressions for the polarizabilities as a function of $Z$ with a relative accuracy of 10${}^{-6}$ up to $Z=100$ are determined by fitting to the calculated polarizabilities. The oscillator strengths satisfy the sum rules ${\sum }_i{f}_{0i}^{(\mathrm{\ell })}=0$ for all multipoles from $\mathrm{\ell }=1$ to $\mathrm{\ell }=4$. The dispersion coefficients for the long-range H-H and H-He${}^{+}$ interactions are given.