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

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”.