Rational spectral transformations and orthogonal polynomials

We show that generic rational transformations of the Stieltjes function with polynomial coefficients (ST) can be presented as a finite superposition of four fundamental elementary transforms: Christoffel transform (CT), Geronimus transform (GT) and forward and backward associated transformations A+T, A−T. It is shown that the Laguerre-Hahn polynomials (LHP) on arbitrary nonuniform lattice are covariant with respect to ST (i.e., ST of a LHP yields another LHP), whereas the semi-classical polynomials are covariant with respect to a subclass of linear ST. Some applications of these results to the theory of the semi-classical polynomials are considered.