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Systems with an O(n) symmetrical Hamiltonian are considered in a$d$-dimensional slab geometry of macroscopic lateral extension and finitethickness $L$ that undergo a continuous bulk phase transition in the limit$L\to\infty$. The effective forces induced by thermal fluctuations at and abovethe bulk critical temperature $T_{c,\infty}$ (thermodynamic Casimir effect) areinvestigated below the upper critical dimension $d^*=4$ by means offield-theoretic renormalization group methods for the case of periodic andspecial-special boundary conditions, where the latter correspond to thecritical enhancement of the surface interactions on both boundary planes. Asshown previously [\textit{Europhys. Lett.} \textbf{75}, 241 (2006)], the zeromodes that are present in Landau theory at $T_{c,\infty}$ make conventionalRG-improved perturbation theory in $4-\epsilon$ dimensions ill-defined. Therevised expansion introduced there is utilized to compute the scaling functionsof the excess free energy and the Casimir force for temperatures$T\geqT_{c,\infty}$ as functions of $\mathsf{L}\equiv L/\xi_\infty$, where$\xi_\infty$ is the bulk correlation length. Scaling functions of the$L$-dependent residual free energy per area are obtained whose $\mathsf{L}\to0$limits are in conformity with previous results for the Casimir amplitudes$\Delta_C$ to $O(\epsilon^{3/2})$ and display a more reasonablesmall-$\mathsf{L}$ behavior inasmuch as they approach the critical value$\Delta_C$ monotonically as $\mathsf{L}\to 0$.

Thermodynamic Casimir effects involving interacting field theories with zero modes