POLYNOMIAL SOLUTIONS OF CERTAIN CLASSES OF ORDINARY DIFFERENTIAL EQUATIONS.

摘要

This thesis investigates generalizations of classical differential equations for which one of the series solutions truncates to a polynomial, the series being the result of the Power Series (or Taylor Series) method of solution.;Various properties of polynomial solutions to (*) are studied; and a generalization of the Hermite polynomials is obtained.;Continuing in this manner, similar results are obtained for the equations of Laguerre, Legendre and Chebyshev.;Finally, comments and questions are presented concerning second order linear differential equations of a more general type.;In particular, the equation y'' (x) - p x('M) y' (x) + p r x('M-1)y (x) = 0, (*) where M is an integer greater than 1, p (NOT=) 0 and r is a positive integer, is considered. This equation can be thought of as a generalization of Hermite's Differential Equation.