q-analogue of de Rham cohomology associated with Jackson integrals, I

In this note we want to give a new formulation of Jackson integrals involved in basic hypergeometric functions through the classical Barnes' representations. We define a q-analogue of de Rham cohomology which can be formulated by means of q-version of Sato's b-functions and derive associated holonomic q-difference system. The evaluation of its multipli- city will be given as a number of different asymptotics.