On the archimedean and nonarchimedean 𝑞-Gevrey orders

Abstract

q q -Difference equations appear in various contexts in mathematics and physics. The “basis” q 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 q -Gevrey order expressing the growth rate of its coefficients : a (nonarchimedean) q − 1 q^{-1} -adic q q -Gevrey order when q q is a parameter, an archimedean q q -Gevrey order when q q is a fixed complex number. The objective of this paper is to relate these two q 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 ) ∈ C ( q ) [ [ z ] ] f(q,z) \in \mathbb {C}(q)[[z]] be a series solution of a linear q q -difference equation, where q q is a parameter, and assume that f ( q , z ) f(q,z) can be specialized at some q = q 0 ∈ C × q=q_{0} \in \mathbb {C}^{\times } of complex norm > 1 >1 . On the one hand, the series f ( q , z ) f(q,z) has a certain q − 1 q^{-1} -adic q q -Gevrey order s q s_{q} . On the other hand, the series f ( q 0 , z ) f(q_{0},z) has a certain archimedean q 0 q_{0} -Gevrey order s q 0 s_{q_{0}} . We prove that s q 0 ≤ s q s_{q_{0}} \leq s_{q} “for most q 0 q_{0} ”. In particular, this shows that if f ( q , z ) f(q,z) has a nonzero (nonarchimedean) q − 1 q^{-1} -adic radius of convergence, then f ( q 0 , z ) f(q_{0},z) has a nonzero archimedean radius converges “for most q 0 q_{0} ”.