Christoffel transform of classical discrete measures and invariance of determinants of classical and classical discrete polynomials

Abstract

Given a finite set of complex numbers U and a classical discrete measure μ, we consider the Christoffel transform ∏u∈U(x−u)μ. Under mild conditions on the finite set U, we show that the orthogonal polynomials with respect to ∏u∈U(x−u)μ enjoy a bunch of non trivial determinantal representations in terms of the classical discrete orthogonal polynomials with respect to μ. Under additional assumptions on the finite set U, we dualize these determinantal representations and find some invariance properties for quasi Casoration determinants whose entries are classical discrete orthogonal polynomials. Passing to the limit we recover some known invariance properties for Wronskian determinant whose entries are classical polynomials.