Ladder operators and differential equations for orthogonal polynomials

Abstract

Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel-Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.