On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi

Brüggemann, Franz

doi:10.1090/S0002-9939-1991-1057941-9

On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi
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Proc. Amer. Math. Soc. 113 (1991), 371-379 Request permission

Abstract:

We prove the following conjecture that is due to Hellerstein and Rossi: Let $\left \{ {{w_1}, \ldots ,{w_n}} \right \}$ be a fundamental system of \[ Lw = {w^{(n)}} + {a_{n - 1}}(z){w^{(n - 1)}} + \cdots + {a_0}(z)w \equiv 0\] with polynomials ${a_j}(z)(0 \leq j \leq n - 1)$. If each ${w_k}(1 \leq k \leq n)$ has only finitely many nonreal zeros, then there exists a polynomial $q(z)$ such that ${u_k}: = \exp (q(z)){w_k}(1 \leq k \leq n)$ form a fundamental system of a homogeneous linear differential equation with constant coefficients.

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