On Barnes' multiple zeta and gamma functions

We show how various known results concerning the Barnes multiple zeta and g amma functions can be obtained as specializations of simple Features shared by a quite extensive class of Functions. The pertinent functions involve L aplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic diff erence equations. Both of these new approaches to the Barnes Functions give rise to novel integral representations. (C) 2000 Academic Press.