author : Louis Brand

The title of this book calls for a word of explanation. In my student days at Harvard I was greatly impressed by a course given by Maxime Bôcher on linear differential equations of the second order in which he pointed out the far-reaching analogies between linear differential and difference equations. It was in this course that he made a remarkable piece of mathematical deduction, for he pointed out that an unpublished (and lost) paper of Charles Sturm, “On the Distribution of Heat in a Series of Vases,” must inevitably have led Sturm to the discovery of his famous theorem on the location of roots in an algebraic equation. The Sturmian sequences used in this theorem are the solutions of certain difference equations.

In this book these analogies have been systematically developed; there is a close correspondence not only between the basic theorems but also in methods of procedure. Thus the methods of “undetermined coefficients” and “variation of parameters” are precisely parallel in both cases. In each case a solution in closed form is possible only when a particular solution of the homogeneous equation is known; and then the general solutions have the same structure.