On the zero-free solutions of linear periodic differential equations in the complex plane
In [J. Reine Angew. Math. 344, 1-21 (1983; Zbl 0524.34007)], \textit{S. Bank} and the reviewer considered the second-order linear differential equation (1) \(y''+A(z)y=0\) for \(A(z)=B(e^z)\), with \(B(\zeta)=K_k\zeta^{-k}+\dots+K_0+\dots+K_l\zeta^l\) with \(K_lK_k\neq 0\). If \(f\) is a nontrivial solution to (1) such that the exponent of convergence \(\lambda(f)\) is the zero-sequence of \(f\) is finite, then \[ f(z)=\psi(e^{z/q})\exp\Bigl(\sum_{j=m'}^md_je^{jz/q}+dz\Bigr), \] with \(q=1\) or \(q=2\), \(d\in\mathbb{C}\) and \(\psi(\zeta)=c_s\zeta^s+\dots+c_0\) with simple zeros only. If \(q=2\), then \(d_j=0\) for all even \(j\). The present paper offers an interesting continued analysis of the situation described above. Recalling that \(f\) is a zero-free solution if and only if \(\lambda(f)=0\), let \(l\) be odd, \(l\geq|k|\geq 0\) and suppose that (1) admits a zero-free solution \(f\). Then \(k=0\), \(K_1=\dots=K_{l-1}=0\), \(K_0=-\frac{l^2}{16}\) and \(f(z)=\exp(d_le^{lz/2}-lz/4)\), where \(l^2d_l^2+4K_l=0\). Moreover, if \(l\) is even, then \(k\) is even as well. If \(l\geq k\geq 0\) we set \(\zeta=e^{z/2}\), and define \(B_0(\zeta)\) by \(B(\zeta^2)=B_0(\zeta^2)+o(\zeta^{l-k})\). Suppose again that \(f\) is zero-free. Then \(f'(z)/f(z)=R(e^{z/2})\), where \(R(\zeta)\) is rational with poles of multiplicities \(l\) at \(\infty\), resp. \(k\) at \(0\), determined by the Laurent series of \(cB_0(\zeta^2)^{1/2}-\zeta^2B_0'(\zeta^2)/4B_0(\zeta^2)\) around \(\zeta=\infty\). These two theorems are typical examples of the results contained in this paper. Moreover, examples are given to illustrate aspects of the results obtained. As a final remark, we observe that the present paper complements the papers [Complex Variables, Theory Appl. 23, No. 1-2, 101-121 (1993; Zbl 0801.34006) and Czech. Math. J. 44, No. 1, 91-107 (1994; Zbl 0805.34032)] by \textit{S. Bank} where a method was proposed to solve some periodic linear differential equations with few zeros only.
second-order linear differential equation
scientific article; zbMATH DE number 1563149