On the explicit determination of certain solutions of periodic differential equations of higher order
In [Complex Variables, Theory Appl. 23, No. 1-2, 101-121 (1993; Zbl 0801.34006)], \textit{S. Bank} developed a method to test a linear differential equation \(y''+ A(z)y= 0\), \(A(z)\) a non-constant periodic entire function which is a rational function of \(e^z\), for the existence of solutions \(f\) satisfying \(\lambda(f)< \infty\), and to specify these solutions. In this paper, the method will be extended and simplified to cover the class of linear differential equations \[ y^{(k)}+ \sum^{k- 2}_{\nu= 1} A_\nu y^{(\nu)}+ A_0(z) y= 0,\tag{\(*\)} \] where \(A_1,\dots, A_{k- 2}\) are complex constants, \(k\geq 3\), and \(A_0(z)\) is a non-constant periodic entire function which is a rational function of \(e^z\). In fact, the idea of S. Bank to determine approximate square roots of a rational function will be extended to determine approximate \(k\)th roots. Then \(k^2\) linear differential equations will be constructed in such a way that (1) if none of these equations possesses a non-trivial polynomial solution, then \((*)\) possesses no non-trivial solution \(f\) such that \(\log^+ N(r, {1\over f})= o(r)\) and (2) whenever anyone of the \(k^2\) linear differential equations possesses a non-trivial polynomial equation, then this polynomial determines a solution \(f\) of \((*)\) such that the exponent of convergence for its zero-sequence \(\lambda(f)< \infty\).
scientific article; zbMATH DE number 872697
scientific article