The second-order electron self-energy in hydrogen-like ions

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A calculation of the simplest part of the second-order electron self-energy (loop after loop irreducible contribution) for hydrogen-like ions with nuclear charge numbers 3 ≤ Z ≤ 92 is presented. This serves as a test for the more complicated second-order self-energy parts (loop inside loop and crossed loop contributions) for heavy one-electron ions. Our results are in strong disagreement with recent calculations of Mallampalli and Sapirstein for low Z values but are compatible with the two known terms of the analytical Zα-expansion.