  In this note we revisit one of the first known examples of exceptional
orthogonal polynomials that was 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. This characterization also leads to a family of general orthogonal
polynomials with missing degrees and this approach manifests its relation to
biorthogonal polynomials introduced by Iserles and N{\o}rsett, which are
applicable to a whole range of problems in computational and applied analysis.
We also obtain a modification of the Christoffel formula for this family since
its classical form cannot be applied in this case.
