System of equations for the vertex functions of the $T$-exponential function

An operator-valued functional of the “cutoff parameters” is constructed; after a certain formal passage to the limit this goes over into the renormaltzed perturbation series for the $S$-matrix of a scalar field with the interactions$\lambda\,{:}\varphi^3{:}$, $\lambda\,{:}\varphi^4{:}$. The characteristic equation obtained for it enables one to construct a system of equations that do not contain ultraviolet divergences for the vertex functions of the $\Sigma_n$ contributions from all strongly connected diagrams with $n$ external lines.