Two-Orthogonal Polynomial Sequences as Eigenfunctions of a Third-Order Differential Operator

This paper presents functional identities fulfilled by the forms of the dual sequence of polynomial eigenfunctions of certain differential operators, belonging to the class of the two-orthogonal polynomial sequences. For a specific third-order lowering operator, the correspondent matrix differential identity is deduced, proving that the resultant polynomial sequence is a classical polynomial sequence in the Hahn's sense. As an example, the vectorial relation fulfilled by the tuple of functionals (u (0), u (1)) of a two-orthogonal polynomial sequences analogous to the classical Laguerre polynomials is given, treated in a work of Ben Cheikh and Douak.