On the asymptotics of the ratio of orthogonal polynomials

Conditions are obtained for the existence of “exterior” asymptotics for orthogonal polynomials. In particular, it is shown that if $\rho'>0$ almost everywhere on the interval $[-1,1]$ ($\rho(x)$ is a nondecreasing function on $[-1,1]$), then for the corresponding orthonormal polynomials the relation $\frac{P_{n+1}(z)}{P_n(z)}\rightrightarrows z+\sqrt{z^2-1}$ holds on compact subsets of $\mathbf C\setminus[-1,1]$. The branch of the square root is chosen so that $|z+\sqrt{z^2-1}\,|>1$ in the region described.
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