Universal Relations in Asymptotic Formulas for Orthogonal Polynomials

Orthogonal polynomials $$P_{n}(\lambda)$$ are oscillating functions of $$n$$ as $$n\to\infty$$ for $$\lambda$$ in the absolutely continuous spectrum corresponding Jacobi operator $$J$$ . We show that, irrespective any specific assumptions on coefficients , amplitude and phase factors asymptotic formulas linked by certain universal relations found paper. Our proofs rely study a time-dependent evolution generated suitable