On the archimedean and nonarchimedean $q$-Gevrey orders

Abstract:

$q$-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 $q_{0}$”.