### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 085, 16 pages      arXiv:1706.02391      https://doi.org/10.3842/SIGMA.2017.085
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### The Inverse Spectral Problem for Jacobi-Type Pencils

Sergey M. Zagorodnyuk
School of Mathematics and Computer Sciences, V.N. Karazin Kharkiv National University, Svobody Square 4, Kharkiv 61022, Ukraine

Received June 10, 2017, in final form October 24, 2017; Published online October 28, 2017

Abstract
In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil $J_5 - \lambda J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. In the case of a special perturbation of orthogonal polynomials on a finite interval the corresponding spectral function takes an explicit form.

Key words: operator pencil; recurrence relation; orthogonal polynomials; spectral function.

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