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We consider a new class of perturbation expansions, which incorporate in asystematic way the available information about the divergent character of theperturbation series in QCD. The new expansion functions, which replace thepowers of the coupling, are defined by the analytic continuation in the Borelplane, achieved through an optimal conformal mapping. We consider severalpossibilities of implementing the known behaviour of the Borel transform nearthe leading singularities in the complex plane and using the correspondingexpansions. These expansions have the same asymptotic convergence rate, butdiffer at low orders. We show that the new expansions allow a precisedetermination of the strong coupling $\alpha_s$ from the hadronic decays of the$\tau$ lepton.

New perturbation expansions in quantum chromodynamics and the determination of $\alpha_s$