A pair of difference differential equations of Euler-Cauchy type

Abstract

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which mul- tiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here (Theorem 11) that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplica- tive constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with re- tarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.