On differential relations of 2-orthogonal polynomials

A generic differential operator on the vectorial space of polynomial functions was presented in a recent work and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Using the techniques therein developed, we prove an identity fulfilled by different differential operators and apply it in a systematic approach to the problem of finding polynomial eigenfunctions, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. In particular, we analyse a third order differential operator that does not increase the degree of polynomials.