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A relativistic description of the structure of heavy alkali atoms andalkali-like ions using S-spinors and L-spinors has been developed. The corewavefunction is defined by a Dirac-Fock calculation using an S-spinors basis.The S-spinor basis is then supplemented by a large set of L-spinors for thecalculation of the valence wavefunction in a frozen-core model. The numericalstability of the L-spinor approach is demonstrated by computing the energiesand decay rates of several low-lying hydrogen eigenstates, along with thepolarizabilities of a $Z=60$ hydrogenic ion. The approach is then applied tocalculate the dynamic polarizabilities of the $5s$, $4d$ and $5p$ states ofSr$^+$. The magic wavelengths at which the Stark shifts between different pairsof transitions are zero are computed. Determination of the magic wavelengthsfor the $5s \to 4d_{\frac32}$ and $5s \to 4d_{\frac52}$ transitions near$417$~nm (near the wavelength for the $5s \to 5p_j$ transitions) would allow adetermination of the oscillator strength ratio for the $5s \to 5p_{\frac12}$and $5s \to 5p_{\frac32}$ transitions.

Relativistic calculations of quasi-one-electron atoms and ions using Laguerre and Slater spinors