Invariance properties of Wronskian type determinants of classical and classical discrete orthogonal polynomials

Abstract

Given a finite set F={f1,⋯,fk} of nonnegative integers (written in increasing order of magnitude) and a classical discrete family (pn)n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casorati determinant det⁡(pfi(x+j−1))i,j=1,⋯,k. In this paper we prove an invariance property of this kind of Casorati determinants when the set F is substituted by the set I(F)={0,1,2,⋯,max⁡F}∖{max⁡F−f:f∈F}. Our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators (Krall discrete polynomials). These polynomials are orthogonal with respect to certain Christoffel transforms of the classical discrete measures. By passing to the limit, this invariance property is extended to Wronskian type determinants whose entries are Hermite, Laguerre and Jacobi polynomials.