  This paper is devoted to the negative flows of the derivative nonlinear
Schr\"odinger hierarchy (DNLSH). The main result of this work is the functional
representation of the extended DNLSH, composed of both positive (classical) and
negative flows. We derive a finite set of functional equations, constructed by
means of the Miwa's shifts, which contains all equations of the hierarchy.
Using the obtained functional representation we convert the nonlocal equations
of the negative subhierarchy into local ones of higher order, derive the
generating function of the conservation laws and the N-soliton solutions for
the extended DNLSH under non-vanishing boundary conditions.
