Characterizations of classical orthogonal polynomials

ABSTRACT. We reconsider the problem of classifying all classical orthogo-nal polynomial sequences which are solutions to a second-order differential equation of the form ‘2.x/y00.x/C ‘1.x/y0.x / D n y.x/: We first obtain new (algebraic) necessary and sufficient conditions on the coef-ficients ‘1.x / and ‘2.x / for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classifi-cation of all classical orthogonal polynomials: up to a real linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi. 1