Darboux transformations and self-similarity for orthogonal polynomials and biorthogonal rational functions

Alex Zhedanov

Abstract We discuss general theory of the Darboux (i.e. Christoffel and Geronimus) transformations for orthogonal polynomials and biorthogonal rational functions. The classical and semiclassical orthogonal polynomials are characterized by some self-similarity closure conditions upon the chain of the Darboux transformations. We consider explicit examples connected with semi-classical polynomials of a discrete variable. These examples lead naturally to a "discrete-discrete" version of the famous Laguerre-Freund non-linear difference relations for the recurrence coefficients of orthogonal polynomials.