Symmetries for Casorati determinants of classical discrete orthogonal polynomials

Abstract

Given a classical discrete family ( p n ) n (p_n)_n of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn) and the set of numbers m + i − 1 m+i-1 , i = 1 , ⋯ , k i=1,\cdots ,k and k , m ≥ 0 k,m\ge 0 , we consider the k × k k\times k Casorati determinants det ( ( p n + j − 1 ( m + i − 1 ) ) i , j = 1 k ) \det ((p_{n+j-1}(m+i-1))_{i,j=1}^k) , n ≥ 0 n\ge 0 . In this paper, we conjecture a nice symmetry for these Casorati determinants and prove it for the cases k ≥ 0 , m = 0 , 1 k\ge 0, m=0,1 and m ≥ 0 , k = 0 , 1 m\ge 0, k=0,1 . This symmetry is related to the existence of higher order difference equations for the orthogonal polynomials with respect to certain Christoffel transforms of the classical discrete measures. Other symmetry will be conjectured for the Casorati determinants associated to the Meixner and Hahn families and the set of numbers − c + i -c+i , i = 1 , ⋯ , k i=1,\cdots ,k and k , m ≥ 0 k,m\ge 0 .