Orthogonal polynomials P n (λ) are oscillating functions of n as n → ∞ for λ in the absolutely continuous spectrum of the corresponding Jacobi operator J. We show that, irrespective of any specific assumptions on coefficients of the operator J, amplitude and phase factors in asymptotic formulas for P n (λ) are linked by certain universal relations found in the paper.Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials P n (λ). They act from the space L 2 (R) of functions into the space ℓ 2 (Z + ) of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.