## Jost functions and Jost solutions for Jacobi matrices. I: A necessary and sufficient condition for Szegő asymptotics.(English)Zbl 1122.47029

Let $$J=J(\{a_n\},\{b_n\})$$ be a Jacobi matrix with $$a_n\to 1$$, $$b_n\to 0$$. The main result of the present paper concerns the Szegő asymptotics for orthonormal polynomials $$p_n$$, associated with $$J$$. Let $$Q$$ be an at most countable set of points $$z$$ in the unit disk $$\mathbb{D}$$ such that $$z+z^{-1}$$ is an eigenvalue of $$J$$. Then $$z^n p_n(z+z^{-1})$$ converges to a nonzero limit as $$n\to\infty$$ uniformly on compact subsets of $$\mathbb{D}\backslash Q$$ if and only if the $$a$$’s and $$b$$’s obey the following conditions $\sum_{n=1}^\infty | a_n-1| ^2+| b_n| ^2<\infty,$ and both limits $\lim_{n\to\infty} \prod_{j=1}^n a_j, \qquad \lim_{n\to\infty} \sum_{j=1}^n b_j$ exist, and the first one is a positive number. The equivalent conditions are given in terms of the spectral data of $$J$$. The authors show that the Szegő asymptotics holds at $$z_0\in\mathbb{D}\backslash Q$$ if and only if the Jost asymptotics does at the same point.
[Part II has appeared in Int. Math. Res. Not. 2006, No. 5, Art. ID 19396 (2006; Zbl 1122.47030), see the following review.]

### MSC:

 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 33C47 Other special orthogonal polynomials and functions 39A70 Difference operators

Zbl 1122.47030
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### References:

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