On the zeros of solutions of certain linear differential equations (Q1921419)

It is well known that the order \(\rho\) of a transcendental entire function of finite order is an integer whenever the linear differential equation \(w'' + A(z) w = 0\) has linearly independent solutions \(f_1\), \(f_2\) for each of which the convergence exponent of the zero-sequence satisfies \(\lambda (f_j) < \rho\), see [Trans. Am. Math. Soc. 273, 351-363 (1982; Zbl 0505.34026)]. In this article, an analogue of this fact for higher order equations will be proved. In fact, let \(A_0, \dots, A_{k - 2}\), \(k \geq 2\), be entire and \(A_0\) transcendental of finite order \(\rho\). Moreover, suppose there exists \(\delta > 0\) such that \[ \log \bigl |A_j (z) \bigr |\leq (1 - \delta) {k - j - 1 \over k} \log^+ \bigl |A_0 (z) \bigr |+ O \bigl( \log |z |\bigr) \] for all \(j\) with \(1 \leq j \leq k - 2\) and \(z\) sufficiently large. Let \(f_1, \dots, f_k\) be linearly independent solutions of \(w^{(k)} + \sum^{k - 2}_{j = 0} A_j (z) w^{(j)} = 0\), each satisfying \(\lambda (f_j) < \infty\). Then for \(E = f_1 \cdots f_k\) we have \(2m (r, {1 \over E}) = (k - 1) T(r, A_0) + O (\log r)\) as \(r \to \infty\). Therefore, if \(\lambda (E) < \rho\) or \(N (r, {1 \over E}) = S(r, A_0)\), then \(\rho\) is a positive integer and both \(A_0\) and \(E\) have order and lower order \(\rho\). It remains open whether \(\rho\) has to be a positive integer provided that \(\lambda (E) < \infty\) holds, which has been a well known conjecture for \(k = 2\).