Gell-Mann-Low equation: Relation between the vacuum polarization coefficients up to the eighth order

Gorishny, Kataev, Larin, and Surguladze have obtained the fourth-order Gell-Mann-Low function ψ₄ and the Callan-Symanzik function β₄. In earlier papers, we used their result to establish relationships between the fifth-, sixth-, and seventh-order vacuum polarization coefficients b₅, b₆, and b₇ and the fifth-, sixth-, and seventh-order Gell-Mann-Low functions ψ₅, ψ₆, and ψ₇. In this paper, we summarize the results of the previous papers and extend the calculations to include the relationship between the eighth-order b₈ and ψ₈ using the Gell-Mann-Low equation and other relations for the Gell-Mann-Low function.