On oscillation, continuation, and asymptotic expansions of solutions of linear differential equations (Q1177651)

The author considers the problem of the extension to a higher-order equation of the following classical result due to Hille: For any second order differential equation \(w"+P(z)w'+Q(z)w=0\) where \(P(z)\) and \(Q(z)\) are polynomials, there exist finitely many rays, \(\arg z=\varphi_ j\) for \(j=1,2,\ldots,m\), with the property that for any \(\varepsilon>0\), all but finitely many zeros of any solution \(f\not\equiv 0\) must lie in the union of the sectors \(|\arg z-\varphi_ j|<\varepsilon\) for any \(j=1,1,\ldots,m\). A similar question has been studied by the author in some previous papers, in the case of the higher-order differential equation \(w^{(n)}+R_{n-1}(z)w^{(n-1)}+\cdots+R_ 0(z)w=0\). In the present paper the author shows some results when \(R_ 0\), \(R_ 1,\ldots,R_{n-1}\) belong to a logarithmic differential field of rank zero over \(F(a,b)\) where \(F(a,b)\) is a neighborhood system suitably defined.