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.