  The present paper essentially contains two results that generalize and
improve some of the constructions of [arXiv:0801.1493]. First of all, in the
case of one derivation, we prove that the parameterized Galois theory for
difference equations constructed in [arXiv:0801.1493] can be descended from a
differentially closed to an algebraically closed field. In the second part of
the paper, we show that the theory can be applied to deformations of q-series,
to study the differential dependency with respect to x\frac{d}{dx} and
q\frac{d}{dq}. We show that the parameterized difference Galois group (with
respect to a convenient derivation defined in the text) of the Jacobi Theta
function can be considered as the Galoisian counterpart of the heat equation.
