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We consider the on-shell mass and wave function renormalization constants$Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a secondnon-zero quark mass. We obtain analytic results in terms of harmonicpolylogarithms and iterated integrals with the additional letters$\sqrt{1-\tau^2}$ and $\sqrt{1-\tau^2}/\tau$ which extends the findings fromRef. [1] where only numerical expressions are presented. Furthermore, weprovide terms of order ${\cal O}(\epsilon^2)$ and ${\cal O}(\epsilon)$ at two-and three-loop order which are crucial ingrediants for a future four-loopcalculation. Compact results for the expansions around the zero-mass,equal-mass and large-mass cases allow for a fast high-precision numericalevaluation.

Exact results for $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ with two mass scales and up to three loops