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Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to themeasure $\mu$. Let $T$ be a linear operator acting in the linear space ofpolynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for allpolynomial $p$. 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 tomultiplicative constants, where the polynomials $q_n^i$, $n,i\ge 0$, aredefined by $q_n^i(x)=\sum_{j=0}^n\mu_j^is_{n-j}(x)$, and $\mu_j^i$ are certaingeneralized moments of the measure $\mu$. For $T=d/dx$ we recover a Theorem byLeclerc which extends the well-known Karlin and Szeg\H o identities for Hankeldeterminants whose entries are ultraspherical, Laguerre and Hermitepolynomials. For $T=\Delta$, the first order difference operator, we get somevery elegant symmetries for Casorati determinants of classical discreteorthogonal polynomials. We also show that for certain operators $T$, the seconddeterminant above can be rewritten in terms of Selberg type integrals, and thatfor certain operators $T$ and certain families of orthogonal polynomials$(p_n)_n$, one (or both) of these determinants can also be rewritten as theconstant term of certain multivariate Laurent expansions.

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