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

Abstract

Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $\mu$. Let $T$ be a linear operator acting in the linear space of polynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for all polynomial $p$. We then construct a sequence of polynomials $(s_n)_n$, depending on $T$ but not on $\mu$, such that the Wronskian type $n\times n$ determinant $\det \left(T^{i-1}(p_{m+j-1}(x))\right)_{i,j=1}^n$ is equal to the $m\times m$ determinant $\det \left(q^{j-1}_{n+i-1}(x)\right)_{i,j=1}^m$, up to multiplicative constants, where the polynomials $q_n^i$, $n,i\ge 0$, are defined by $q_n^i(x)=\sum_{j=0}^n\mu_j^is_{n-j}(x)$, and $\mu_j^i$ are certain generalized moments of the measure $\mu$. For $T=d/dx$ we recover a Theorem by Leclerc which extends the well-known Karlin and Szeg\H o identities for Hankel determinants whose entries are ultraspherical, Laguerre and Hermite polynomials. For $T=\Delta$, 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 $(p_n)_n$, one (or both) of these determinants can also be rewritten as the constant term of certain multivariate Laurent expansions.