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

Abstract

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.