Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal number, including properties of corresponding sequences orthogonal polynomials. We also present some invariant semi-invariant under such transformations. For instance, show that Nevai class is in question, which not true general, ratio asymptotic still holds outside spectrum symmetric matrix but could include one extra point. principal, these can be iterated and, for example, demonstrate how lead to RII-recurrence relations, turn are related rational functions pencils matrices.