A Pair of Difference Differential Equations of Euler-Cauchy Type

Details A Pair of Difference Differential Equations of Euler-Cauchy Type A Pair of Difference Differential Equations of Euler-Cauchy Type

2003-10-06

arxiv.org/abs/math/0310077v1

Transactions of the American Mathematical Society, Vol. 355 (2003), No. 12, pp. 4985--5002. MR1997592 (2004g:34119)

Mathematics

Classical Analysis and ODEs

18 pages, 1 figure, AMSLaTeX, published electronically at http://www.ams.org/journal-getitem?pii=S0002-9947-03-03223-9

Scientific paper

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple 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 that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded 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.

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