Improvement of the method of diagonal Padé approximants for perturbative series in gauge theories

Recently it has been pointed out that diagonal Pad\'e approximants to truncated perturbative series in gauge theories have the remarkable property of being independent of the choice of the renormalization scale as long as the gauge coupling parameter ${\alpha}(p^2)$ is taken to evolve according to the one-loop renormalization group equation -- i.e., in the large-${\beta}_0$ approximation. In this letter we propose and describe an improvement to the method of diagonal Pad\'e approximants. The improved method results in approximants which are independent of the chosen renormalization scale when ${\alpha}(p^2)$ evolves at any chosen (in principle arbitrary) loop-level.