Higher logarithms and $\epsilon$-poles for the MS-like renormalization prescriptions

We consider a version of dimensional regularization (reduction) in which the dimensionful regularization parameter $\mathrm{\Lambda }$ is in general different from the renormalization scale $\mu$. Then in the scheme analogous to the minimal subtraction the renormalization constants contain $\epsilon$-poles, powers of $\ln \mathrm{\Lambda }\mathrm{/}\mu$, and mixed terms of the structure ${\epsilon }^{-q}{\ln }^p\mathrm{\Lambda }\mathrm{/}\mu$. For the MS-like schemes we present explicit expressions for the coefficients at all these structures which relate them to the coefficients in the renormalization group functions, namely in the $\beta$-function and in the anomalous dimension. In particular, for the pure $\epsilon$-poles we present explicit solutions of the 't~Hooft pole equations. Also we construct simple all-loop expressions for the renormalization constants (also written in terms of the renormalization group functions) which produce all $\epsilon$-poles and logarithms and establish a number of relations between various coefficients at $\epsilon$-poles and logarithms. The results are illustrated by some examples.