On the properties of polynomials satisfying a linear differential equation. I

Abstract

Introduction. Sequences of polynomials such as the Legendre, the Laguerre and the Hermite polynomials appeared in mathematics many years ago, and their properties have been investigated by numerous people. They satisfy simple difference equations, and also are solutions of linear differential equations of second order. The expansion problems (in the complex plane) associated with them are not so old. For the Legendre polynomials the region of convergence was determined by C. Neumann. t More recently the convergence regions for the Laguerre and Hermite polynomials were treated by 0. Volk.$ The paper of Volk considers, more generally, the boundary value problem (in the complex domain) for a second-order linear differential equation, not restricting attention to polynomials. The nth-order equation has since been treated, as a boundary value problem, by L. Bristow. ? Up to the present, however, there has been no general study of the properties of polynomials satisfying a linear differential equation of order higher than two. The present paper has in view such an investigation. There is another aspect to our treatment. In an earlier work we considered the properties of arbitrary sets of polynomials,|| associating with each set a linear differential equation, usually of infinite order. We obtained certain formal properties, whose complete justification required convergence proofs. The present paper deals with these matters for the case of a finite order equation. ?1 is preliminary: we state two theorems of Perron, and prove a corollary that is of use later. ?2 introduces a fundamental differential equation whose polynomial solutions { yn(x) } we investigate, as well as the entire function