Variations around classical orthogonal polynomials. Connected problems (Q1318447)

A new method was proposed to study classical orthogonal polynomials. It is founded on the duality, specially on the use of the dual sequence of a sequence of polynomials. Characterizations of classical orthogonal polynomials were reviewed, in particular from the Hahn's definition. Two new characterizations were discussed. One is a generalization of Hahn's property and the second uses quadratic relations which generalize the relation by \textit{P. J. McCarthy} [Port. Math. 20, 47-52 (1961; Zbl 0188.124)]. The class of semi-classical orthogonal polynomials was studied. The problem of finding normalized sequences whose respective derivative's sequences are orthogonal was discussed. It was shown that each classical polynomial verifies a fourth-order differential equation. Some of the results were already given previously (\textit{P. Maroni} [C. R. Acad. Sci., Paris, Sér. I 313, 209-212 (1991; Zbl 0736.33008)]).