The generalized Bochner condition about classical orthogonal polynomials revisited

We bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k, for some k 1. So, we build those differential equations explicitly. If k = 1, we get the Bochner’s characterization of classical polynomials. With help of the formal computations made in Mathematica, we explicitly give those differential equations for k = 1, 2 and 3 for each family of the classical polynomials. Higher order differential equations can be obtained similarly. © 2005 Elsevier Inc. All rights reserved.