SPECTRAL TRANSFORMATIONS, SELF-SIMILAR REDUCTIONS AND ORTHOGONAL POLYNOMIALS

We study spectral transformations in the theory of orthogonal polynomi als which are similar to Darboux transformations for the Schrodinger e quation. Linear transformations of the Stieltjes function with coeffic ients that are rational in the argument are constructed as iterations of the Christoffel and Geronimus transformations. We describe a charac teristic property of semiclassical orthogonal polynomials (SCOP) on th e uniform and the exponential lattice; namely, that all these polynomi als can be obtained through simple quasi-periodic and q-periodic (self similar) closures of the chain of linear spectral transformations. In the self-similar setting, a characterization of the Laguerre-Hahn pol ynomials on linear and q-linear lattices is obtained by considering ra tional transformations of the Stieltjes function generated by transiti ons to the associated polynomials.