Another Characterization of the Classical Orthogonal Polynomials

Abstract

The classical orthogonal polynomials of Jacobi, Laguerre and Hermite are characterized as the only orthogonal polynomials with a differentiation formula of the form \[ \pi (x)P'_n (x) = \left( {\alpha _n x + \beta _n } \right)P_n (x) + \gamma _n P_{n - 1} (x),\quad n \geqq 1,\] where $\pi (x)$ is a polynomial. If “orthogonal polynomial” is used in the sense of “orthogonal with respect to a function of bounded variation,” then the characterization remains valid if the Bessel polynomials are included in the classical family. This characterization also permits us to verify a conjecture of Karlin and Szegö.