Thanks for your explanation. Could you please kindly show some references or some keywords about the useful conclusion for Laurent polynomial that I can relate to?

Thank you very much for your quick responds but please excuse me as a rookie. I didn’t quite follow you. Could you please give more details about how $[1]((p^2+q^2)+qp(x+x^{-1}))^n=(2\pi)^{-1}\int_{0}^{2\pi}((p^2+q^2)+qp(e^{it}+e^{-it}))^ndt$. Lots of thanks!