On some ultrametric \(q\)-difference equations (Q1943226)

On an ultrametric algebraically closed field \(K\), the authors treat the linear \(q\)-difference equation \[ \sum_{i=0}^{s}A_i(x)(\sigma^i_q f)(x)=B(x), \eqno(1) \] where \(q \in K\), \(0<|q|<1\), \((\sigma_q f)(x)=f(qx)\) and \(B(X)\), \(A_0(X), \ldots, A_s(X)\) (\(s\geq 1\)) are elements of \(K(X)\) such that \(A_0(X)A_s(X)\neq 0\). Denote by \(\mathcal M(K)\) the field of meromorphic functions on \(K\). By means of the ultrametric Nevanlinna theory, they obtain several results on (1), for example the following. If \(f\in \mathcal M(K)\) is a solution of (1), then \(T(r,f)=O(\log r)^2\) as \(r\to\infty\). If \(f\in \mathcal M(K)\setminus K(X)\) is a solution of (1), then \((\log r)^2=O(T(r,f))\) as \(r\to\infty\). These results are generalizations obtained for the complex number field \(\mathbb C\).