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