This paper is divided in two parts. In the first part we consider a convergent q-analog of the divergent Euler series, with q ∈ (0, 1), and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding q-difference equation. In the second part, we work under the assumption q ∈ (1, +∞). In this case, at least four different q-Borel sums of a divergent power series solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.
The present paper essentially contains two results that generalize and improve some of the constructions of [HS08]. First of all, in the case of one derivation, we prove that the parameterized Galois theory for difference equations constructed in [HS08] 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 d dx and q 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.
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