On the associated orthogonal polynomials

摘要

By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985), Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre–Hahn polynomials (Magnus (1984)), is obtained (cf. Belmehdi and Ronveaux (1989), Dini (1989), Ronveaux (1990)).