On the archimedean and nonarchimedean q-Gevrey orders.

q-Difference equations appear in various contexts in mathematics and physics. The "basis" q is sometimes a parameter, sometimes a fixed complex number. In both cases, one classically associates to any series solution of such equations its q-Gevrey order expressing the growth rate of its coefficients : a (nonarchimedean) q^{-1}-adic q-Gevrey order when q is a parameter, an archimedean q-Gevrey order when q is a fixed complex number. The objective of this paper is to relate these two q-Gevrey orders, which may seem unrelated at first glance as they express growth rates with respect to two very different norms. More precisely, let f(q,z) \in \mathbb {C}(q)[[z]] be a series solution of a linear q-difference equation, where q is a parameter, and assume that f(q,z) can be specialized at some q=q_{0} \in \mathbb {C}^{\times } of complex norm >1. On the one hand, the series f(q,z) has a certain q^{-1}-adic q-Gevrey order s_{q}. On the other hand, the series f(q_{0},z) has a certain archimedean q_{0}-Gevrey order s_{q_{0}}. We prove that s_{q_{0}} \leq s_{q} "for most q_{0}". In particular, this shows that if f(q,z) has a nonzero (nonarchimedean) q^{-1}-adic radius of convergence, then f(q_{0},z) has a nonzero archimedean radius converges "for most q_{0}". [ABSTRACT FROM AUTHOR]