  We show how to use the Hopf algebra structure of quantum field theory to
derive nonperturbative results for the short-distance singular sector of a
renormalizable quantum field theory in a simple but generic example. We discuss
renormalized Green functions $G_R(\alpha,L)$ in such circumstances which depend
on a single scale $L=\ln q^2/\mu^2$ and start from an expansion in the scale
$G_R(\alpha,L)=1+\sum_k \gamma_k(\alpha)L^k$. We derive recursion relations
between the $\gamma_k$ which make full use of the renormalization group. We
then show how to determine the Green function by the use of a Mellin transform
on suitable integral kernels. We exhibit our approach in an example for which
we find a functional equation relating weak and strong coupling expansions.

High Energy Physics - Theory;Quantum Algebra