Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case

We study orthogonal polynomials with periodically modulated recurrence coefficients when $0$ lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Tur\'an determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.