In this paper, we show that Dunkl-classical polynomial sequences can be characterized by a differential-difference equation.

We reconsider the problem of classifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form 2 (x)y (x) + 1 (x)y (x) = λ n y(x). We first obtain new (algebraic) necessary and sufficient conditions on the coefficients 1 (x) and 2 (x) for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classification 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.

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