Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

Abstract

Let (pn)n be a sequence of orthogonal polynomials with respect to the measure μ. Let T be a linear operator acting in the linear space of polynomials P and satisfying deg(T(p))=deg(p)−1, for all polynomial p. We then construct a sequence of polynomials (sn)n, depending on T but not on μ, such that the Wronskian type n×n determinant det(Ti−1(pm+j−1(x)))i,j=1n is equal to the m×m determinant det(qn+i−1j−1(x))i,j=1m, up to multiplicative constants, where the polynomials qni, n,i⩾0, are defined by qni(x)=∑j=0nμjisn−j(x), and μji are certain generalized moments of the measure μ. For T=d/dx we recover a theorem by Leclerc which extends the well-known Karlin and Szegő identities for Hankel determinants whose entries are ultraspherical, Laguerre and Hermite polynomials. For T=Δ, the first order difference operator, we get some very elegant symmetries for Casorati determinants of classical discrete orthogonal polynomials. We also show that for certain operators T, the second determinant above can be rewritten in terms of Selberg type integrals, and that for certain operators T and certain families of orthogonal polynomials (pn)n, one (or both) of these determinants can also be rewritten as the constant term of certain multivariate Laurent expansions.