Multiple commutator expansion for the Lamb shift in a strong Coulomb field

A new expression far the electron self-energy in a strong Coulomb field is obtained without expansion in powers of alpha Z (alpha is the fine structure constant, Z is the charge of the nucleus). This expression contains the multiple commutator expansion, which differs from the alpha Z expansion and converges rapidly even for very high Z values. The renormalization procedure requires the proof of a new sum rule. This proof is given explicitly for Z = 1 and implicitly for all Z values for the 1S(1/2) state. The numerical calculation of the Lamb shift for different Z values is performed and comparison with the known results made.