The present paper
contains two results that generalize and improve constructions of Hardouin and
Singer. In the case of a derivation, we prove that the parametrized Galois
theory for difference equations constructed by Hardouin and Singer 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 dependence with respect
to x and q. We show that the parametrized 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.
Keywords
linear difference equations, Galois theory,
hypertranscendence