Chebyshev Sets of Polynomials which Satisfy an Ordinary Differential Equation

Abstract

It has been known for a long time (S. Bochner, Math. Z., (1929)) that the only polynomial sets $\{ {y_n (x)} \}_{n = 0}^\infty $, where $y_n (x)$ is of degree exactly n in x, satisfying a second-order ordinary differential equation \[ Py''_n + Qy'_n + Ry_n = \lambda _n wy_n ,\] where P, Q, R, W are functions of x, and $\lambda _n $ is an appropriate constant, are the Jacobi, Laguerre, Hermite, Bessel polynomials and powers of x. Considerable recent work has substantially enlarged the class of weight functionals available, however. Further those polynomial sets satisfying similar fourth-order differential equations can also be completely classified.