The invention has an object to provide a catalyst for polymerizing or copolymerizing an α-olefin, catalyst constituent thereof, and method of polymerizing α-olefins with the catalyst, for production of α-olefin polymers or copolymers with high hydrogen response, high polymarization reaction rate, high stereoregularity and excellent melt fluidity. The invention discloses a catalyst constituent of the catalyst for polymerizing or copolymerizing an α-olefin, represented by Formula 37 or 38:Si(OR1)3(NR2R3)Formula 37(where in Formula 37, R1 is a hydrocarbon group with 1 to 6 carbon atoms; R2 is a hydrocarbon group with 1 to 12 carbon atoms or hydrogen; and R3 is a hydrocarbon group with 1 to 12 carbon atoms)RNSi(OR1)3Formula 38(where in Formula 38, R1 is a hydrocarbon group with 1 to 6 carbon atoms; and RN is a cyclicl amino group).