Bank-Laine functions with sparse zeros (Q2718957)

A Bank-Laine function is an entire function \(E(z)\) such that \(E'(z)=\pm 1\) at every zero of \(E(z)\). These functions are related to linear complex differential equations: Given \(A(z)\) entire, let \(f_1\), \(f_2\) be linearly independent solutions of (1) \(w''+A(z)w=0\), normalized such that their Wronskian is \(=1\). Then \(E(z)=f_1(z)f_2(z)\) is a Bank-Laine function satisfying \(4A=(E'/E)^2-2E''/E-1/E^2\). Conversely, if \(E(z)\) is Bank-Laine, then \(A\) is entire and \(E\) is the product of two normalized linearly independent solutions of~(1). Basic results and examples of Bank-Laine functions may be found in [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)]. Moreover, \textit{L. Shen} [Proc. Am. Math. Soc. 95, 544-546 (1985; Zbl 0596.30048)] observed that if \((a_n)\) is a complex sequence of distinct points tending to \(\infty\), then there exists a Bank-Laine function with zero-sequence \((a_n)\). However, this function may be of infinite order, even if the exponent of convergence of \((a_n)\) is finite. Non-existence results for such Bank-Laine functions of finite order have been offered in [\textit{S. M. Elzaidi}, Complex Variables, Theory Appl. 38, No. 3, 201-220 (1999)] and the present paper as well. Perhaps the most interesting part of this paper is that one addressing non-trivial examples of Bank-Laine functions [see also \textit{J. K. Langley}, Arch. Math. 71, No. 3, 233-239 (1998; Zbl 0930.30028)]. The following result will be proved: Let \((c_n)\) be a positive sequence tending to \(+\infty\). Then there exists a Bank-Laine function \(E(z)=e^z\prod_{n=1}^\infty(1-z/\alpha_n)\) with \(|\alpha_n|>c_n\) for each~\(n\). Moreover, \(E\) is of order \(=1\), and its zeros have exponent of convergence \(\lambda(E)=0\). Concerning the corresponding equation~(1), \(A\) is transcendental and \(f_1\) has no zeros. The result means that there exist Bank-Laine functions of finite order with arbitrary sparse zero-sequences. Perhaps we should add that a complete understanding of the Bank-Laine functions would also solve the Bank-Laine conjecture which asks whether \(\max(\lambda(f_1),\lambda(f_2))=\infty\) provided \(f_1\), \(f_2\) are linearly independent solutions of~(1) with \(A(z)\) entire of non-integer finite order.