Proceedings of the American Mathematical Society, 136 (2008), 2803-2814. 
Abstract:
We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since deg_q and ord_q define two valuations on C(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition.
doi:10.1090/S0002-9939-08-09352-0
Arxiv:0709.2464.
HAL:hal-00350715v1