Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders.

In the paper, we make a detailed study on the generalized Crewther Relation (GCR) between the Adler function ($D$) and the Gross-Llewellyn Smith sum rules coefficient ($C^{\rm GLS}$) by using the newly suggested single-scale approach of the principle of maximum conformality (PMC). The resultant GCR is scheme-independent, whose residual scale dependence due to unknown higher-order terms are highly suppressed. Thus a precise test of QCD theory without renormalization scheme and scale ambiguities can be achieved by comparing with the data. Moreover, a demonstration of the scheme independence of commensurate scale relation up to all orders has been presented. And as the first time, the Pade approximation approach has been adopted for estimating the unknown $5_{\rm th}$-loop contributions from the known four-loop perturbative series.