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