  We present a new approach to the static finite temperature correlation
functions of the Heisenberg chain based on functional equations. An
inhomogeneous generalization of the n-site density operator is considered. The
lattice path integral formulation with a finite but arbitrary Trotter number
allows to derive a set of discrete functional equations with respect to the
spectral parameters. We show that these equations yield a unique
characterization of the density operator. Our functional equations are a
discrete version of the reduced q-Knizhnik-Zamolodchikov equations which played
a central role in the study of the zero temperature case. As a natural result,
and independent of the arguments given by Jimbo, Miwa, and Smirnov (2009) we
prove that the inhomogeneous finite temperature correlation functions have the
same remarkable structure as for zero temperature: they are a sum of products
of nearest-neighbor correlators.
