Let \((p_n)_n\) be a classical discrete family of orthogonal polynomials. For a finite set \(F=\{f_1,\ldots,f_k\}\) of nonnegative integers the Casorati determinant is the determinant of the form \[ \det(p_{f_i}(x+j-1))_{i,j=0,\ldots,k}, \quad f_i\in F, \ \ i=0,\ldots,k. \] For example, if \((p_n)_n\) is the family of Charlier polynomials, then the above, denoted by \(C^a_{F,x}\), is called the Casorati-Charlier determinant.
The authors prove some invariance properties for the Casorati determinant for Charlier, Meixner and Hahn polynomials \((p_n)_n\). Taking limits the authors also obtain some invariance properties for Wronskian type determinants for Hermite, Laguerre and Jacobi polynomials. For example, the authors show that if the family of Charlier polynomials \((p_n)_n\) is “normalized by taking its leading coefficient equal to \(\frac{1}{n!}\)”, then for each \(a\not=0\) and a finite set \(F\) of nonnegative integers the following holds \[ C^a_{F,x}=(-1)^{w_F} C^{-a}_{I(F),-x}, \] where \(w_F\) is the degree of \(C^a_{F,x}\) (as a polynomial of \(x\)) and \[ I(F)=\{1,2,\ldots, \max F\}\setminus\{\max F-f:\, f\in F\}. \]