The discrete spectrum of Jacobi matrix related to recurrence relations with periodic coefficients

TL;DR

This paper investigates the discrete spectrum of Jacobi matrices associated with recurrence relations having periodic coefficients, focusing on specific cases like N=3 and N=4, and introducing related polynomial families.

Contribution

It provides new analysis of the discrete spectrum for Jacobi matrices with periodic coefficients, including special cases and polynomial families introduced by the authors.

Findings

Discrete spectrum characterized for N=3,4,5 cases

Introduction of elementary N-symmetrical Chebyshev polynomials

Analysis of Jacobi matrices with periodic recurrence coefficients

Abstract

In this note we investigate the discrete spectrum of Jacobi matrix corresponding to polynomials defined by recurrence relations with periodic coefficients. As examples we consider a)the case when period $N$ of coefficients of recurrence relations equals three (as a particular case we consider "parametric" Chebyshev polynomials introduced by authors early); b)the elementary $N$-symmetrical Chebyshev polynomials ($N=3,4,5$), that was introduced by authors in the study of the "composite model of generalized oscillator".