Journal of the
Korean Mathematical Society JKMS

We reconsider the problem of classifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form $$\ell_2(x)y''(x)+\ell_1(x)y'(x)=\lambda_ny(x).$$ We first obtain new (algebraic) necessary and sufficient conditions on the coefficients $\ell_1(x)$ and $\ell_2(x)$ 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 $\italic real$ linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi.

Keywords: classical orthogonal polynomials, second-order differential equations

MSC numbers: Primary 33A65