Universal relations in asymptotic formulas for orthogonal polynomials

Abstract

Orthogonal polynomials $P_n(\lambda )$ are oscillating functions of $n$ as $n\to \infty$ for $\lambda$ 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 $P_n(\lambda )$ 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 $P_n(\lambda )$. They act from the space $L^2(\mathbb{R})$ of functions into the space ${\ell }^2({\mathbb{Z}}_{+})$ of sequences. We consider such operators in a rather general setting and find necessary and sufficient conditions of their boundedness.

References in corpus (2)

• Asymptotic behavior of orthogonal polynomials without the Carleman condition
• Scattering theory for Laguerre operators