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Elaborating on a previous letter, we use a new approach to compute energylevels of a non-relativistic bound-state of two constituents, with masses m andM, by systematic expansions - one in powers of m/M and another in powers of(1-m/M). Technical aspects of the calculations are described in detail.Theoretical predictions are given for O(alpha(Z*alpha)^5) radiative recoil andO((Z*alpha)^6) pure recoil corrections to the average energy shift andhyperfine splitting relevant for hydrogen, muonic hydrogen, and muonium.Comment: 9 pages, revte

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Expansion of bound-state energies in powers of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>M</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mn /><mml:mi>−</mml:mi><mml:mi>m</mml:mi><mml:mo>/</mml:mo><mml:mi>M</mml:mi><mml:mo>)</mml:mo></mml:math>

Expansion of bound-state energies in powers ofm/Mand(1−m/M)