with Charlotte Hardouin. Pacific Journal of Mathematics, Vol. 256 (2012), No. 1, 79--104. 
Abstract:
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 xd/dx and qd/dq. 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.
ArXiv:1103.5067
HAL:hal-00794743