DEK-Type exceptional orthogonal polynomials and a modification of the Christoffel formula

In this note we revisit the polynomials introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. We then use this characterization to show how to construct a family of exceptional orthogonal polynomials from any family of orthogonal polynomials. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.Comment: 20 page