Solutions with exponential character to a linear functional equation and roots of its characteristic equation

We study the connection between solutions of the functional equation ϕ(x) = S ϕ(x + M(s))σ (ds) which are comparable at infinity with functions of the form xeλx, and real roots of the characteristic equation S eλM(s)σ(ds) = 1. © 2006 Elsevier Inc. All rights reserved.