The recoil, vacuum polarization and electron vertex corrections of first and second orders in the fine structure constant $alpha$ and the ratio of electron to muon and electron to alpha-particle masses are calculated in the hyperfine splitting of the
$1s^{(e)}_{1/2}2s^{(mu)}_{1/2}$ state of muonic helium atom (mu e ^4_2He) on the basis of a perturbation theory. We obtain total result for the muonically excited state hyperfine splitting $Delta u^{hfs}=4295.66$ MHz which improves previous calculations due to the account of new corrections and more accurate treatment of the electron vertex contribution.
We present new investigation of the Lamb shift (2P_{1/2}-2S_{1/2}) in muonic deuterium (mu d) atom using the three-dimensional quasipotential method in quantum electrodynamics. The vacuum polarization, nuclear structure and recoil effects are calcula
ted with the account of contributions of orders alpha^3, alpha^4, alpha^5 and alpha^6. The results are compared with earlier performed calculations. The obtained numerical value of the Lamb shift 202.4139 meV can be considered as a reliable estimate for the comparison with forthcoming experimental data.
On the basis of the perturbation theory in the fine structure constant $alpha$ and the ratio of the electron to muon masses we calculate one-loop vacuum polarization and electron vertex corrections and the nuclear structure corrections to the hyperfi
ne splitting of the ground state of muonic helium atom $(mu e ^3_2He)$. We obtain total result for the ground state hyperfine splitting $Delta u^{hfs}=4166.471$ MHz which improves the previous calculation of Lakdawala and Mohr due to the account of new corrections of orders $alpha^5$ and $alpha^6$. The remaining difference between our theoretical result and experimental value of the hyperfine splitting lies in the range of theoretical and experimental errors and requires the subsequent investigation of higher order corrections.
